It is a common perception that Darwin's theory of evolution is as well-supported a scientific theory as Newton's theory of gravity or Einstein's theory of relativity. Though evolutionary theory has changed much since Darwin's time, it's fundamental weaknesses, which Darwin himself recognized, remain unexplained today by evolutionary biologists.
Before any of the evidence is discussed, however, it should be noted that evolutionary theory can NEVER be as well-supported as Newton's theory of gravity or Einstein's theory of relativity simply because evolutionary theory is a historical science of biological origins. This is not a fault of evolution per se, for historical sciences investigate unrepeatable events of the past. Just like a historian trying to reconstruct the reasons for the fall of the Roman Empire, evolutionary biologists deal with theories trying to account for past events of evolutionary history. They look at evidence and make inferences about what they hypothesize happened. There can be no experiments, such as those showing that apples always fall to the ground or that the speed of light is constant in all reference frames, which can prove evolution happened.
Many scientists say that although evolution isn't proven, that it's still the "best" theory we have to explain the origins of life-forms on earth. These sorts of statements keep evolution alive and well in secular academia (which the public has typically trusted as objective and unbiased in their methodologies). However, these assertions deny a host of counter-evidences showing that evolution didn't occur, and cover up for a purely philosophical definition of science which excludes the possibility that there was an intelligent designing Creator a priori. Much as we might expect, evolutionary theory shows that theories which don't start with God, don't end with God, and run contrary to the observed data. The following is a brief qualitative discussion of some of the most common types of evidences against the theory of evolution:
Darwin himself realized that there were certain types of biological structures which Darwinian evolution simply couldn't create. Darwinian evolution works by a process called the mutation-selection mechanism. Purely random mutations in the DNA create new types of organisms, and those organisms which are better at surviving and reproducing tend to leave more offspring. Over time, populations of organisms change as those which are best suited to the environment get "selected" (i.e. they survive and reproduce better). The catch to all of this is that changes must occur at a very slow pace, one little mutation at a time. Also, biological systems produced by evolution must be functional (i.e. confer some benefit to the organism) at every little step along their evolutionary path. For this reason, Darwin said:
"If it could be demonstrated that any complex organ existed which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down."
(Charles Darwin, Origin of Species)
Molecular biologist Michael Behe recently noted in his book, "Darwin's Black Box", that there are a host of biochemical systems which function much like machines. These machines work only if all the necessary parts are present for if one part is removed, the entire machine breaks down. These systems defy an evolutionary origin, because they cannot be built up in a step-by-step manner. Examples Behe gives include the bacterial flagellum, the blood clotting mechanism, and the biochemical processes behind vision. Reverse-engineering of these and other biological structures shows that evolutionary processes weren't at work in creating them.
William Dembski, in "The Design Inference" and "No Free Lunch" shows that Darwinian processes cannot explain much of biological complexity. Biological organisms--both at the macro and micro level--exhibit this sort of "irreducible complexity" and "specified, complex information. These properties of biological systems cannot be explained by evolution.
In the 19 th century, long before the discovery of DNA, Gregory Mendel realized that biological traits are passed down in an orderly manner, which today we know happens through DNA, chromosomes, and genes. Darwin's theory predicts that if all organisms share a common ancestor, then patterns of genes will reflect the line-of-descent and biologists should be able to construct well-ordered family trees going all the way back to a common ancestor for all living organisms simply by looking at the genetic makeup of an organism. However, comparisons of different DNA sequences and gene distributions often yield evolutionary trees which contradict one-another, meaning that there is no grand "tree of life" which can be constructed showing that all organisms are interrelated. In fact, the best explanation for this phenomenon is that there was a common "design" behind these organisms by a "designer", not "common descent." Molecular biologist Jonathan Wells’ in "Icons of Evolution" or molecular biologist Michael Denton in, "Evolution: A Theory in Crisis" both give good explanations of this problem.
Many who took evolution in school probably heard the phrase, "ontogeny recapitulates phylogeny", meaning that evolutionary history is reflected during the growth and develpoment of an organism. A common evidence for this is the alleged presence of fish gills in human embryos during growth. Though humans do indeed have gill-like structures (very different from gills--they're more like small wrinkles in the neck) this classic evidence for evolution has been debunked as it has been shown that it's originator, Ernst Haeckel, faked the data. In fact, comparisons of embryos of humans, fish, chickens, and amphibians show that organisms develop in a way NOT predicted by evolutionary theory. These embryos begin very different, briefly become somewhat similar at an intermediate stage, and then end very different. If ontogeny really does recapitulate phylogeny, and these organisms have a common ancestry, embryos should be similar from the very beginning and grow more different during development. This, however, is not the case, making this a powerful evidence against common ancestry. Again, Jonathan Wells’ book, "Icons of Evolution" is a good reference for this evidence.
It is a common misperception that "natural selection" or "speciation" (macroevolution) are all there is to evolutionary theory. If "natural selection" or "speciation" are true, then somehow people assume that evolution works fine to explain everything. In fact, natural selection is only half of the mechanism behind evolutionary theory, and evidence for "speciation" often depends only on the definition of terms which biologists use.
Natural selection is a fact, but all it really means is non-random death. In fact, it is a mere mathematical certainty given variation in a species and "selection pressure" fromthe environment. But before natural selection can become important, there first must be variation upon which selection can act. The problem for evolution lies in the mutation-selection mechanism, where random mutations--the ultimate originator of all variation--must somehow account for the origin of the vast and often irreducible complexity (mentioned above) of life on earth. Natural selection would still give us no answer to answer to the question "how did the different shades of moths originate in the first place?"
Speciation is simply a term evolutionary biologists use to describe the processes by which new species evolve in the wild--though it is often equated with "macroevolution". However, when evolutionists claim evidence for "speciation", they have often only witnessed trivial microevolutionary changes in two very similar populations.
A good example is "Darwin's Finches" in the Galapagos Islands. Darwin found that different islands had different finch populations, and concluded that all the finches were descended from a single finch population which migrated to the Islands long ago. Because these finches live on different islands, they are "reproductively isolated" (a typical definition for a species), and we call them separate species. But have we really witnessed any radical evolution here? The differences between the finch "species" are almost trivial--the main difference lies only in slight changes in the size of their finch beaks, and the finches can still interbreed in captivity. Evidences like these do not tell us how important macrevolutionary transitions took place--such as the evolution of beaks, wings, feathers, or humans--in the first place. In fact, these testify that even after many many generations of reproductive isolation, populations tend to change very little, providing evidence against evolution of very different species.
Perhaps the classic argument against evolutionary theory comes from the fossilized remains of organisms found to have existed throughout time. If purely natural evolution has actually occurred in the past and all living organisms share a common ancestor then one ought to find the fossilized transitions between one form and another in the fossil record. However, it has been well known since Darwin's time that plausible "transitional fossils" rarely exist:
"... The number of intermediate varieties, which have formerly existed on the earth, (must) be truly enormous. Why then is not every geological formation and every stratum full of such intermediate links? Geology assuredly does not reveal any such finely graduated organic chain; and this, perhaps, is the most obvious and gravest objection which can be urged against my theory." (Darwin, C. (1859) The Origin of Species (Reprint of the first edition) Avenel Books, Crown Publishers, New York, 1979, p. 292)
Out of tens of thousands of species known from the fossil record, only a few are claimed to be Darwin's missing transitional forms. However, a close analysis of these few fossils (commonly cited ones are Archaeopteryx (a bird), Ambulocetus (a land mammal), and Acanthostega (an amphibian)) reveal that they do not shed any light on the origin of the important features of their respective groups. Harvard paleontologist Stephen J. Gould said,
" The absence of fossil evidence for intermediary stages between major transitions in organic design, indeed our inability, even in our imagination, to construct functional intermediates in many cases, has been a persistent and nagging problem for gradualistic accounts of evolution." (Stephen J. Gould, 'Is a new and general theory of evolution emerging?' Paleobiology, vol 6(1), January 1980, pg 127)
Gould proposed a theory called "Punctuated Equilibrium" which is meant to account for the lack of transitional forms, saying we don’t find transitional forms because transitions didn’t have a chance to be fossilized. But punctuated equilibrium does not fit with the workings of genetics--too much biological complexity must be built with far too few rolls of the dice. The lack of transitional forms remains unaccounted for and is a strong evidence against evolution. See Phillip Johnson’s, "Darwin on Trial" for a good account of problems with the fossil record.