The Genealogy of Jesus is utterly important to the early Christians. Matthew and Luke spend dozens of verses to show that Jesus comes from the house of David. These Genealogies were meant to provide proof because it was (and remains) a well known fact that in order to be a Messiah one has to be a son of David.
Therefore, these proofs had better hold up to close scrutiny because they are literally the lynchpin of the entire New Testament.
The Genealogies do not match what is written in Chronicles. They are not accurate.
It is a well-known fact that under Jewish Law as decreed by God in Numbers 1:18 cf. Numbers 36:7 and many other places. This is not being picayune; the law clearly states (again, these are God's decrees not Rabbinic fences) that tribal inheritance will not pass between tribes. This means your tribe cannot change because as a landed Israelite you owned a plot of land in Israel. With this kind of connection to the land your tribe could never change. This was the purpose and meaning of the separation into Tribes, why it was Patrilineal.
Moreover this is admitted by many Christian Apologists – it is not an unknown problem. For example Josh McDowell acknowledges this issue (and quotes Jeremiah 22:30) on page 188 of “A Ready Defense”. Also, the Ryrie Study Bible (page 1315) also acknowledges the issue but claims the curse had no effect since this was “from Mary's genealogy” – a claim which makes no sense even if it were true (it's not; see 'conflicting genealogies' below).
Also note that even Martin Luther conceeds on this issue; in 1543 he writes, “it cannot be proved that Jesus is of the tribe of Judah, because the apostle Matthew traces the tribe of Judah to Joseph and not Mary; therefore he could not be the Messiah as proved by the tribe of Judah and David.' Martin Luther does note that Mary could be of the house of David; yet again this disqualifies the genealogy based on the adoption principle (and going through Nathan vs. Solomon; see below, 'cursed genealogies').
There appears to be a major problem with the accounts in Matthew and Luke. They don't match. It actually appears there are two different genealogies despite it being stated they are both through Joseph.
The two conflicting genealogies given for Jesus are those of Matthew 1 and Luke 3.
The claim is sometimes made that one genealogy is of Mary. However, in Matthew 1:16, Matthew's genealogy is stated to be through Joseph – and in Luke 3:23, Luke's genealogy is stated to be through Joseph. So the claim that one genealogy is Mary's seems to be an invention out of whole cloth.
It is also interesting to point out that even if the above two points could somehow be resolved, both genealogies would still be invalid – that is, they would actually show the exact opposite – that Jesus cannot be the Messiah!
In Matthew 1:12, the genealogy passes through Jeconiah. The issue here is that Jeconiah's descendants were cursed and his descendants can no longer hold the office of King:
In Luke 3:31, we see the genealogy passed through Nathan. However, this again amounts to a disproof of Jesus' Messiahship, according to 1 Chronicles 22:9-10!
The Genealogies are not accurate (conflicting), don't even apply (adoption) and even if they did apply they only serve to show Jesus could not have been the Messiah.