Origin of the Genetic Code
The deepest mystery in biology, and in science generally, is the origin of life and the genetic code. Life as we know it cannot exist without DNA and the information it stores. But DNA and the genetic code cannot have come before life because it takes a vast and intricate system of molecular machines just to reproduce DNA and translate it into the proteins that do the actual business of metabolism and constitute the machinery of life, including all the machinery to turn DNA into protein. It's the chicken and egg problem in it's sharpest molecular form.
DNA is not translated directly into protein. First it must be transcribed into messenger RNA, which in turn is translated into protein by a complex apparatus that consists of proteins and other kinds of RNA molecules. This fact is probably one element that suggested that maybe RNA was the molecule that came before gene and protein. Proteins are the most versatile components of our molecular machinery, acting as structural elements and tiny molecular machines of the most versatile sort, but their really big role is chemical catalysis - promoting chemical reactions. It turns out, however, that RNA can also do some catalytic trickery.
The basic notion of RNA world is that before modern life (like bacteria) evolved 3 billion years or so ago, there must have been an RNA world where RNA molecules somehow collaborated to catalyze their own replication and other simple chemical reactions which could have powered them. Nobody (so far as I know) can yet explain in any detail how this could have worked, but the hypothesis has gained credibility from recent discoveries that show that RNA has many more functions in the modern cell than were hinted at in textbooks not so long ago. Much of the DNA in cells is not, in fact, translated into protein, but into RNA, only some of the functions of which we understand. In any case, it's clear that it's still busy in crucial catalytic functions.
So maybe the real ancient ones aren't DNA molecules, but RNA molecules, which a few billion years ago invented DNA to propagate themselves, in the process destroying most fossil evidence of the world that came before.