Despite substantial progress in understanding the mechanisms of carcinogenesis and fighting oncology diseases, cancer mortality remains rather high. Therefore, there is a striving to reduce this mortality to the level determined by endogenous biological factors. The review analyzes the mutations that lead to cell malignant transformation and describes the contribution that self-renewal of adult tissues makes to tumorigenesis. Cancer progression is considered as a development of a complicated system where cells mutate, evolve, and are subject to selection. Cancer paradoxes are described in conclusion.