My personal problem with reincarnation is the issue of personal identity. When body dies, what makes you you dies too, irregardless of whether you've got a new body at the end of it.
... does it? Or, more explicitly, does it
entirely? People
are somewhat prepositioned towards certain behavioral patterns, even outside of environmental influences -- definitely not
much, in most cases not involving radical brain structure abnormalities, but somewhat. It's also entirely
possible that the process of reincarnation (somehow) leaves you in a body/situation that will produce a person very similar (not identical, of course -- most forms of reincarnation I've seen do posit some sort of change between incarnations) to your last incarnation. A sort of, "Round Two, FIGHT" scenario involving characters that randomize slightly every round but are still largely the same competitor. Incidentally, I think I'd totally play a fighting game where you weren't entirely sure how many limbs you'd have this match.
And @MH, there's various disciplines and whatnot for "connecting" with/learning about previous lives, with varying degrees of "success". Almost certainly just hallucinations -- sometimes rather vivid and/or self-induced -- at best, buuuuut many of the people that believe they've experienced the phenomena think it's real enough. Considering there's very few ways to prove or disprove their claims,* and almost certainly no way to do so with certainty for
all of their claims, well... E: Well, you mostly just pat them on the back and let them believe what they believe, at least until they start digging up ancient buried treasure or something.
Badly off topic, though. I know there's at least
some christian denominations that believe in reincarnation to one degree or another, but I have no clue about their doctrinal or theological stances. Mostly just that they exist, and at least one believes in
limited reincarnation that occurs until... whatever their rapture equivalent is.
*You'd have to have someone who believes they lived a past life (and remember some/all of it) that was thoroughly documented enough someone could check their details, and then find some way to confirm the person in question had never encountered said documentation -- and even if they get it all
right, coincidentally accurate WAG is a possibility. And getting it wrong could just be inconsistent narration -- the
documentation being wrong.
Documentation of that sort is almost entirely nonexistent, currently. Another three or four hundred years and maintenance of facebook archives or whatever and that might change, but it's pretty impossible at th'mo.