A common battle cry from creationists who dispute the very existence of evolution by natural selection is that evolution, even if it can explain differences in the lengths of bones and muscles and so forth, can't explain increases in complexity occasioned by, for example, the ability of an organism to produce a chemical enzyme for which it previously had no structure to produce that enzyme.
Here's how it all went down: on February 24, 1988, twelve isolated populations of the common bacteria E. coli were placed in test tubes containing a glucose nutrient medium. Over the twenty-one years since the experiment began, the nutrients have been replenished every day (the website of the project amusingly describes the "inexorable" routine that the project adheres to), and over forty-thousand generations of E. coli have consequently been spawned (E. Coli reproduces very quickly). All of the colonies have undergone changes at the genetic level which allow them to consume glucose more efficiently, and at the same time have lost the ability to efficiently consume other sugars which E. coli generally consumes, but which are not available in this experimental environment.
Now it just so happens that there was a bit of another potential nutrient called citrate in those test tubes, generated partly by the breakdown of the mitochondria in the dead E. coli cells themselves. E. coli normally has no mechanism that is able to process citrate in order to turn it into energy, but here the conditions were such that the ability to process citrate would be beneficial. And indeed one of the twelve colonies underwent mutations which enabled it it process citrate as well as glucose -- a clear increase in the complexity of the organism!! This mechanism was not introduced all in one stroke. Instead, the record of the evolution of these bacteria (of which samples were regularly taken for exactly this sort of testing) shows that what happened was that a mutation occurred here and there which was neither helpful nor harmful, but which introduced a piece of the mechanism that could break down citrate. And when at last a mutation occurred which led to the introduction of a means to tie together the pieces of the mechanism as part of the organism's larger process of consuming the materials around it, well, there they were to be tied together.
The theory of evolution does not propose that evolution occurs by some grand plan where a particular design is the inevitable outcome. It is a series of flukes -- the particular mutation observed in the E. coli experiment has only happened in one of the twelve samples despite tens of thousands of cycles of reproduction being engaged in by billions of bacteria, and even though it would be beneficial if it were to happen in all of the colonies. Maybe it had never happened anywhere before, and will never happen anywhere again. But it does not need to, for it has already proved beyond rational question is that mutations can occur that increase complexity. In short, it proves that evolution can explain all introductions of additional complexity, as long as those introductions can be broken down into pieces to be carried along in the organism's junk DNA, and not by themselves drive the organism to its death.