The theory of evolution has problems that cannot be resolved using current scientific understanding.

Abiogenesis: Non-living compounds somehow assemble to form simple, self-replicating structures. These structures then evolve to form primitive cells. The process is simply assumed to have occured, nobody has explained exactly how the process happened, and what the odds of it were. Even after the formation of a protocell, the evolution of each organelle is a major challenge, as I shall illustrate in my next point.Macro-evolution: It is assumed that microevolutions accumulated over time and caused the creation of different species. However, this process, like the process of abiogenesis, is simply assumed to have occured. Nobody can explain exactly how it all happened, nor has anyone calculated the odds of such occurances. One example is the evolution of eye. It is believed that eyes first evolved as patches of photosensitive cells, which gave the organism some survival advantage. Later, a depression formed in the area, giving the the organism the ability to sense the direction of light. This depression increased with time, and gradually the eye took the form of a pinhole camera. Note that at each stage the mutation is fairly simple, and could have occured by chance. This theory is widely accepted today by scientists, but I find it lacking sense. The appearance of photosensitive cells is a fairly simple mutation, but does not give any survival advantage unless the cells can send signals to other parts of the organism and those other parts are capable of deriving information out of such signals. Thus for these primitive "eye-patches" to get selected, a complicated apparatus has to be evolved along with it in one full go, or else it will provide no survival benefit, and will not be selected for. Also, once the eye-spots have developed (along with the whole mechanism to derive useful information from it and to respond adequetely), the formation of depressions alone will not give any survival advantage unless the organism has programs that can derive additional information regarding the direction of light from the signals sent by the photocells. Thus, if one tries to explain macro-evolution as gradual accumulation of microevolutions, one cannot do that. For each "micro" change to be selected for, it must provide a survival advantage. When a new organ is evolved, it is first supposed to come into existence as a very simple structure (say a proto-organ), and then gradually develop its features. However, if an additional organ has to develop, (even if it is a simple proto-organ) it must also form pathways of communication with some other organs, and those other organs must also learn to understand the signals that this new organ gives, and how to respond to them. Only then will this new "organ"(or a primitive structure) give a survival advantage and be selected for. This process is rarely feasable, as we have seen in the above example. In short, the arising of a primtive organ or a modification in an existing organ could be explained as a chance mutation. However, if this new modification has to give a survival advantage, it must make changes in the entire system. Such changes are unlikely to occur in a single mutation, and must proceed thorugh a series of mutations (with each stage providing a survival advantage to be selected). However, none of the intermediary stages would give suffiecient survival advantage to be selected for. For example, the mere appearance of a patch of photocells does not give any survival advantage because the entire system still lacks a mechanism to receive signals from the eye-patch and also lacks the requisite programming to interpret those signals and responding to them.Lack of probability theory: No version of the theory of evolution is able to calculate the odds of life arising out of non-living matter, and subsequently evolving into complex life forms. This amounts to saying that there is no mathematical proof that the proposed timeline of evolution contains enough time for evolution to occur. There will be enough time for evolution to occur if the odds are good, but if the odds are very high, even trillions of years might not suffice to produce the amount of complexity that we see today. I'm not saying evolution did not occur. I'm merely saying that the theory currently a lacks a mathematical basis in probability theory. Without any proper mathematical basis, the process is simply assumed to have occured, just because scientists cannot come up with a better explanation than natural selection.The Cambrian explosion: the relatively rapid appearance, around 530 million years ago, of most major animal phyla, as demonstrated in the fossil record, accompanied by major diversification of organisms including animals, phytoplankton, and calcimicrobes. Before about 580 million years ago, most organisms were simple, composed of individual cells occasionally organized into colonies. Over the following 70 or 80 million years the rate of evolution accelerated by an order of magnitude (as defined in terms of the extinction and origination rate of species) and the diversity of life began to resemble that of today.