I have never been able to resolve (what seems to me to be) the contradiction between the theory of evolution and the 2nd Law of Thermodynamics, or entropy.
Entropy (simply stated because I'm not even remotely a scientist) is the natural and inevitable movement of things from an ordered state to a disordered state. Entropy has been called "time's arrow," in that it establishes what things will be in the future (and why traveling backwards in time is impossible).
The theory of evolution, on the other hand, suggests that things (in this case, biological entities) become naturally and inevitably more complex over time...that is, creatures evolve from single cell to multi-cell organisms, for example.
I have also been intrigued by Michael Behe's concept of "irreducible complexity."
Further, I do not know that there has ever been a documented case of evolution in a currently living species. Species are becoming extinct all of the time (part of natural selection) and we continue to discover new species but there just does not seem to be any evidence of an existing species undergoing evolution. It seems that there would be and that it would be a quantifiable situation.
Geological records, on the other hand, are not really interpretable any other way.
Pretty cool paradox, eh?
EDIT:
And, opening up yet another can of worms, it seems to me that natural selection and the theory of evolution would demand that homosexuality would be an evolutionary dead end and would have ceased to exist as an inherited trait thousands of years ago. I find this amusing because on the one hand you have those who insist that homosexuality is strictly behavioral (i.e. a lifestyle choice) and on the other hand you have those (usually militant homosexual advocates) who insist that it is in their genes and they have no choice in the matter whatsoever.
Curiously, the militant homosexual advocates seem to come down on the side of supporting evolution I suppose because it eliminates the presence of a deity (and the inevitable moral consequences thereof) from their lives. And yet, it also seems that evolutionary science would tend to dictate that homosexuality cannot and should not exist. Further, with the advent of DNA decoding and genetic mapping, militant homosexuals seem to be, mostly, extremely opposed to the identification of a "homosexual gene." I have heard that their opposition stems from the notion that they would be able to be "identified" through genetic testing, effectively outing them, and also that prospective parents might genetically engineer any trace of homosexuality out of their future children.
The left is quite fond of saying how science can be quite troubling when it gets in the way of one's dogma and yet they have no trouble whatsoever ignoring that same thought when they run up against it.