  We review the major progress in the rigorous analysis of the classical
quasispecies model that usually comes in two related but different forms: the
Eigen model and the Crow--Kimura model. The model itself was formulated almost
50 years ago, and in its stationary form represents an easy to formulate
eigenvalue problem. Notwithstanding the simplicity of the problem statement, we
still lack full understanding of the behavior of the mean population fitness
and the quasispecies distribution for an arbitrary fitness landscape. Our main
goal in this review is two-fold: First, to highlight a number of impressive
mathematical results, including some of the recent ones, which pertain to the
mathematical development of the quasispecies theory. Second, to emphasize that,
despite these 50 years of vigorous research, there are still very natural both
biological and mathematical questions that remain to be addressed within the
quasispecies framework. Our hope is that at least some of the approaches we
review in this text can be of help for anyone embarking on further analysis of
the quasispecies model.
