Most people do not really think to ask this question about Evolution. However, it has become quite the important question in determining the validity of the paradigm. In this post when I refer to "evolution" I am referring to "macroevolution" (see my post "How Did It All Happen? Part 4- Evolution? Really?"). If I am talking about microevolution (see same post above), I will make the distinction.
As I discussed in the previously cited post, random mutation does happen, and natural selection does operate on those mutations. This observation has been extrapolated into the theory of Evolution. According to the paradigm life began as a single-cell organism, and through the process noted we arrive at the state of life today (complex, mega-multicellular organisms).
Most familiar with biology are familiar with the Long Term Evolution Experiment (LTEE). This experiment is touted as demonstrating that mutation does take place and can be observed. This demonstration is then used to conclude that Evolution is correct as a theory. Unfortunately, the LTEE actually raises a huge problem for the Evolutionary paradigm. The issue of historical contingency.
Noted evolutionary biologist Stephen Jay Gould describes Evolution as being unrepeatable. He used the analogy of a cassette tape. He stated that if the history of life was a tape, and that tape were rewound and played back, the second play would be radically different from the first.
The LTEE has demonstrated that random mutations that result in a usable feature will only repeat if the feature is only one mutation away from being produced. And the frequency of that repeat was extremely small in the experiment. This means that random mutation can only produce the same outcome twice if the final mutation is a single step away and the chances of that final mutation being the right mutation are minuscule.
So, what's the big deal? Well, we find many instances of "evolution repeating itself" in nature. Probably the most easy to notice is a feature of some animals, know as the wing. The closest "distant evolutionary relatives" that both have wings are bats and birds. According to the evolutionary paradigm, the lineages that lead to the evolution of bats and birds diverged long before wings ever evolved. Since they are so far removed from each other, it is safe to say that the final mutation that lead to the complete and functional wing was more than a single step away.
This poses the problem of nature contradicting what is demonstrated in the lab. If the theory states that evolution can repeat itself, then the LTEE provides powerful evidence that either that assertion is false or not enough time has transpired since life first began until now for life to reach the complexity that it has. If the theory states that evolution cannot repeat itself, then nature has proven it incorrect. Keep in mind that I am still referring to "evolution" here as "macroevolution". The LTEE does establish "microevolution", but falsifies a key component of "macroevolution". If the macroevolutionary paradigm is to survive this new discovery, it will need to be changed. The change would be that evolution can repeat itself, but only after more generations than the LTEE produced (but less than the smallest difference between two repeats that took place) or once a certain defined threshold is reached. Either way, as long as the LTEE continues the new detail added to the evolutionary theory can be tested.
For more details about the experiment's results and their implications please read the article by biochemist Dr. Fazale Rana entitled "Inability to Repeat the Pass Dooms Evolution". Dr. Rana includes links in this article to other articles that he has written regarding the issue.
I also recommend the podcast I Didn't Know That where you can ask Dr. Rana questions.
Check out Wikipedia's list of 80+ examples of evolution repeating itself beyond a single mutation.