A brief explanation:

• Modern chronology is based primarily on the birth in Matthew under Herod the Great and the massacre of the innocents in Bethlehem (about two years before Herod's death in 4 BC), the 15th year of Tiberius and the age of Jesus according to Luke, and the astronomically calculated years (30 and 33) for the Jewish Passover on Thursday (the day of the week in Mark, Matthew, and Luke, but not in John). Modern chronology has three variants:
  
  • 29/30. An appearance by Jesus in the year 29/30 is preferred by those who interpret the 3 and 14 years in Paul's letter to the Galatians as 17 years between his conversion in Damascus and an apostolic council before the long journey in Acts in which Paul is led before Governor Gallio (Acts 18:12). 
  • 27/30. Because the year 30 leaves very little time for the appearances of John and Jesus (and therefore for the story of the Gospels), some argue for a different interpretation of the 15th year of Tiberius, two years earlier than usual in antiquity. This gives a three-year appearance of Jesus in the years 27 to 30.
  • 30/33. Those who wish to adhere to the usual dating of Tiberius' reign and to a longer mission opt for a three-year ministry in the years 30 to 33; this interpretation of Paul and Acts is somewhat less compelling.

Modern chronology implies a major error in Luke's chronology, inconsistent with the author's self-presentation as a historian. Because of this, many scholars do not want to reconstruct Paul's life on the basis of Acts but solely on the reading of his letters (especially Galatians). This would eliminate the data points for the arrest of Peter, Paul before Gallio and Paul before Felix and Festus.

Astronomy provides a false accuracy because the Jewish calendar is not astronomically fixed (it requires human decisions about the start of months and the intercalation of extra months to align the lunar months with solar years).

My biggest problem with modern chronology is that it leaves very little time for the missions of John the Baptist and Jesus, while introducing a period of about ten years in which very little happened. The variant in which the 15th year of Tiberius is dated earlier to 26/27 (to allow for an earlier start of John the Baptist) runs contrary to coinage of Tiberius.

• The author of Luke and Acts used governors, kings, and emperors to set the chronology of his story. Luke did not use a uniform calendar. We must therefore approach his chronology based on the dates we have reconstructed from the rulers he mentions, the age of Jesus, and the implied duration of certain activities in his story. This is possible, provided you don't use the reference to Herod the Great in Matthew. Read this way, you would conclude that Luke has Jesus born when Quirinius takes over the administration from Herod the Ethnarch (the son of Herod the Great, also known as Archelaus). The crucifixion in Luke would then be placed as late as possible in Pilate's term of office (26-36; note that Josephus also mentions the crucifixion as Pilate's last act in Judea), when Luke's Jesus is in his thirtieth year of life.
• The Church Fathers who interpreted the data several centuries later were only familiar with the dates of the emperors. The dates for kings and governors were not readily available. They calculated the year of birth based on the 15th year of Tiberius and the age of Jesus (Luke 3), understood as 30 years shortly after John the Baptist began. The year of the crucifixion was derived from the duration of his ministry (three Passovers). Eusebius of Caesarea speaks of a journey by Peter to Rome in the second year of Claudius.
• The monk Dionysius Exiguus, who calculated the Anno Domini back in the early seventh century, did not make any calculation errors, as is often claimed. He used a different method and different starting points. He calculated backwards from the Christian system of determining of the annual date for Easter (developed in Alexandria).in order to get the most symbolically significant times for the crucifixion and birth of Jesus. He preferred a date for the birth of Jesus when the days were getting longer again. The date of birth for John the Baptist then falls on June 24, six months earlier (Luke 1:36), when the days are getting shorter again. After all, John said of Jesus: "He must increase, I must decrease" (John 3:30).