  The history of the quadratic stochastic operators can be traced back to work
of S.Bernshtein (1924). During more than 80 years this theory developed and
many papers were published. In recent years it has again become of interest in
connection with numerous applications to many branches of mathematics, biology
and physics. But most results of the theory were published in non English
journals, full text of which are not accessible. In this paper we give a brief
description of the results and discuss several open problems.
