The Year of the Birth of Christ - late 6 - mid 5 BC
The modern calendar counts years from the birth of Jesus. But Jesus' birth-year was miscalculated to 1BC/1AD. This was probably because of the statements in Luke that John the Baptist began preaching, with Jesus shortly after, in the 15th year of Tiberius (28-29 AD) and that Jesus was around 30 when he did (Luke 3:23). Jesus' birthdate is neither 1 BC nor 1 AD (nor the non-existent 0 AD), but is in fact a few years before 1 AD. The most common date given for the year of Jesus' nativity is 4BC. This is because of the mention of his birth while Herod was alive (Matthew 2:1), sometime shortly before his death (Matthew 2:19; in v.20 Jesus is still presumably an infant).
We can see from the statement in Matthew 2:7, 16 that it is said that Jesus was born at least a year and some months (less than 2 years according to v.16) prior to the Magi's visit. It's not said how long after the journey to Egypt Herod died, but the text implies that it wasn't too long; perhaps a few months. In conclusion, we can assume Jesus was around 1.5-2 years old at the time of Herod's death.
Ultimately however, the date of Herod's death needs to be determined. According to conventional dating following Josephus, he died between a lunar eclipse (March 13, 4 BC) and the Passover (April 12). There remains little doubt that Herod probably did die in 4BC, as opposed to any other date. Josephus himself dates it to 4BC on numerous occasions, and Herod's sons dated their reigns from 4BC. However, it has been long recognized (e.g. William Whiston in his translation) that the time given is too short for the events mentioned prior to Herod's death and after the eclipse. It is therefore probably best to put Herod's death in late 4BC. This may explain why Appian places Herod's accession de jure in 39BC instead of 40BC as does Josephus (since Herod ruled 36 years, 37 in Josephus as per Jewish counting, and the possible closeness to 3BC, may have made Appian count 36 years back from 3BC to get 39BC). Moreover, the dating of Herod's sons seems to necessitate that Herod died no earlier than September, 4 BC.[Ken Doig, New Testament Chronology]
The 15th Year of Tiberius
The chronology of Jesus in Luke 3:23 makes him to be "about 30 years" in the 15th year of Tiberius. There are two things to discuss here. The first year of Tiberius would have certainly been reckoned from 14 AD when he became emperor, disregarding any coregencies or de jure claims for an earlier date ("co-princeps" with Augustus in 12 AD). [Suetonius, Tiberius 21] This is how his reign would have been commonly dated by anyone as Josephus does [Antiquities 18.4.6; 18.6.10], regardless of locale or regional calendars, even though they continued to use the month names of the land (e.g. in Egypt, Egyptian months: "4th year of Hadrian, Pharmouthi 27" [Loeb Classical Library (L266), C. C. Edgar, Select Papyri Vol. 1 (1970), p.207]). This would be especially true decades after when only the universal Roman reckoning would've remained recognizable for a Roman emperor (Josephus uses Jewish reckoning for Judean kings and Roman for Roman emperors).
It's true that some manuscripts have Josephus date the death of Philip to Tiberius' 22nd year instead of 20th in Antiquities 18.4.6, which, if authentic, could be interpreted either that the event took place two years later (35/36 instead of 33/34), or that Tiberius' reign was reckoned from 12 AD instead of 14. But later, in 18.6.10, Josephus clearly assigns a beginning date for Tiberius' reign by saying he reigned 22 years, 5 months, 3 days (a few weeks short), which means, since he died in early 37 AD, he must've counted his first year from 14 AD. Suetonius does the same. [Tiberius 73]
"About" 30 Years of Age
The statement, "about 30" is, like the entire verse, in accordance with how the Greek writers described someone. [e.g. in contracts, Loeb Classical Library ibid] It could be argued that Luke intended Jesus' age to be exactly 30, despite the relative description, "about", because documents used the same phrase to describe specific numbers (e.g. "Apollonius, aged about 26 years, of medium height, with fair complexion, long face, and a scar on the right foot..." [Loeb Classical Library, ibid., p.37]; "Psennesis also called Krouris son of Horus, Persian of the Epigone, of the village of Gotnit in the lower toparchy of the Latopolite nome, aged about 45 years, of medium height or under, dark-skinned, rather curly-haired, long-faced, straight-nosed, with a scar on the under lip..." [p.83]).
But these documents were written in present times by personal family members or close acquaintances. And they still approximated, unlike other timekeeping (e.g. "for 16th months...in Caesar's 17th year" [p.47]; a boat leased for 60 years [p.113], or a papyrus marsh for 3 years [p.125]). This is because of the minor amount of uncertainty that can exist about the age of an individual that doesn't need to relate in a legally-binding document.
Sometimes the person's age was unknown ("aged over 30" [p.251]), sometimes for brevity's sake ("[they] who are minors" [pp.103, 105]; cf the adoption of a son who is "about 2 years old" [p.31]), and sometimes the person stated the age without the "about" [pp.165, 249] which shows these are indeed approximations regardless of their almost universally specific nature. These were exceptions, so it's clear that despite the specific numbers found in the vast majority of private contracts, these were still approximations and Luke used it as such. So we don't need to suppose Jesus must've been exactly 30 years old in Tiberius' 15th year.
In conclusion, I think late 6 to mid 5 BC is the best estimate for the year of Christ's birth. No other year is within the boundary of both the historical and biblical evidence.
The Star of Bethlehem
Closely tied to the year of Jesus' birth is the star of Bethlehem which the Magi saw. There have been many suggestions for what astronomical event it could have been, close to 4 BC, if we take it as a natural event. Was it a natural or supernatural cosmological phenomenon? Let's start with the question of the Magi.
If they were converts, they must've been recent ones: they don't seem to know of Herod's dangerous character, which would've been found out soon after they made a few pilgrimages to Jerusalem for the feasts. In Parthia, it's possible they knew few enough Jews to only know some particulars of the religion, without much about political specifics in Judea itself. So this rules them out being priests or any other high ranking Jewish religious officials either; these had to help them locate the birthplace of the Messiah in Bethlehem (Matt. 2:5-6). It's almost as if they saw the star, got ready, and left, because certainly there would've been enough educated Jews in Parthia to inform them of these things, but possibly they only asked for the way to Jerusalem, thinking that's where it led (Bethlehem is very close and from Parthia would've seemed the star pointed to the capital). Their description of him as "the king of the Jews," would've been a natural development from the star's significance; possibly they thought it was a newborn son of Herod's. They find an alternate route out of Bethlehem (MT 2:12) probably through the guidance of locals; not that anyone in Jerusalem would've recognized them anyway.
How this star and its meaning was revealed to them can't be figured out for sure. Possibly part of some personal, private revelation or understanding, or else all Parthian Magi would've come. Perhaps the main one of them had a dream (cf. Matt. 2:12). Or maybe some other event, natural (astronomical), or otherwise event (prophet?), got them thinking. Maybe the turbulence at the time surrounding the Parthian throne caused these righteous Zoroastrian priests to inquire of and be subsequently given the identity of the real "Shahan" (Persian royal title, meaning "King of Kings") was.
The star reappears once they leave Jerusalem. Perhaps they assumed it was leading them to the capital and didn't bother checking on it too much and had lost track of it, or simply didn't look too closely. Possibly it had been daytime, hence no stars, when they arrived in the vicinity of Jerusalem and Bethlehem, which are so close to each other, for this assumption to be mixed with not knowing at all. Upon understanding that the Messiah would be from Bethlehem, they left and it reappeared at night. This star leading them to a specific house would suggest that it wasn't an ordinary star, or maybe the farms in Bethlehem were so few and spaced out that some astronomical table could actually lead them to a specific vicinity of one of them. Would this really be the case if they were previously so off that they went to Jerusalem? Maybe they started verifying more closely after they dropped their assumptions, or there was a more specific movement of the star after.
Herod probably trusted the unsuspecting Magi enough and didn't consider it a very serious threat for a child-king to challenge him so he didn't bother having them followed to the nearby village. But, as we know from Josephus, he would've become paranoid enough over the mere fact that there were reports of a "king of the Jews" somewhere so close, so he could've decided to cover his bases and kill the children 2 and younger; he certainly didn't bother finding out if anyone had left, so he was probably trying to kill a ridiculous legend, knowing the possible propaganda of a child-king could be just as powerful as if there actually were one. This incident must've caused some ruckus at the court for it to be remembered at least in such vague but specific terms.
This also suggests that the star was a regular one in the night sky and there was nothing special in its physical appearance. We can see that no one besides the Magi knew what it look like, needing to ask them when it appeared. The second appearance of the Star of Bethlehem was similarly indistinguishable and "incognito" from the other stars as the first (Matthew 2:9) because it would've caught Herod's attention long before the escape of the Magi or Jesus' family.
Since the star was no special astronomical phenomenon, we can exclude comets and supernovae. Special star conjunctions are a possibility, but that can't be determined. So it wasn't Haley's comet in 12 BC, the comet of 10 BC, or the supernovae of 5 and 4 BC. Matt. 2:10 states the star reappeared, so if this indistinguishable for the untrained eye star lasted for at least several weeks (up to 2 years), from the departure of the Magi to reaching Jerusalem, it would've been nothing other than an astronomically and mathematically decipherable connection or conjunction. Specifically the triple conjunction of 7 BC (September 15), would be a bit too early, but it's not impossible: this was also soon after the census of Augustus (8 BC), which would reach the provinces and give local rulers an excuse to tax non-Romans (as Josephus notes about Herod) a little later. It would be possible for Luke to round Jesus' age to 30, when Jesus would've been 34.
The month of Jesus' birth
If we are completely unsure of the year of Jesus' birth, how much more of the month, or the date. Even so, again there are subtle factors that give relative periods. The timeless objection that the shepherds could not have had flocks out on the field during winter (Luke 2:8) is probably true. The verse doesn't seem to indicate anything but that their flocks were outside in the field, and a winter night would be very improbable for this.
Chronology of the infancy of Christ
The star would have certainly appeared when Jesus was born, not conceived: in Matt. 2:2 they ask about where was one who was born. It does not make much sense to have the announcement at conception as opposed to birth. Also, Herod calculates the age of the children he slaughters from when the star was first seen, implying the Magi told him it signified his birth. Clearly Jesus and his family stayed in Bethlehem for a little over a year while the census was going on and Mary was recovering; perhaps they liked it there better and Joseph could've had work.
The year of the start of Christ's ministry
There are three markers for the year Jesus began his ministry. The first is the mention of the Temple construction for 46 years in John 2:20. We know from Josephus that this was begun some time after Passover of 20 BC, and certainly the project could not have taken off until late 20 BC. Based on coin evidence at the stone quarry, archaeological evidence suggests it started in 19 BC. From late 20BC/early 19BC, 46 years brings us to early 28 AD, and the text in John suggests this was the first Passover of Jesus' ministry.
Second, the start of John the Baptist's ministry is given as the 15th year of Tiberius (Luke 3:1). The text (Luke 3:1-18,21) does not necessitate a period of more than a month or two, and in fact seems to imply that Jesus was amongst the first to be baptized by John the Baptist. The 15th year of Tiberius would be Sept. 28-29. Despite different calendars, Luke would've used the conventional, universal understanding of the term like other papyri of the time.
Third, Luke 3:23 states that Jesus was "about thirty years" when he started his ministry. As was mentioned, he could've been a couple of years older than 30. It's doubtful Luke knew the actual year of Jesus' birth: he would've simply known a rough average of Jesus' age at the start of his preaching, which he could narrow down by correlating with John the Baptist's career. In fact, the chronological marker is directly about John the Baptist and not Jesus. So what he must've done was looked up information on when the Baptist preached, which wasn't for too long, and in that way cleverly related it to Jesus' beginning: the 15th year of Tiberius (28/29AD; Luke 3:1).
The length of the ministry of Jesus
The year of the crucifixion