Recognition of finite groups by the set of element orders and related problems

A finite group $G$ is said to be recognizable by the set of element orders if $G$ is uniquely, up to isomorphism, defined by this set in the class of finite groups. For example, the alternating group of degree 5 is recognizable since it is the unique finite group whose set of element orders is equal to $\{1,2,3,5\}$. More generally, the problem of recognition by the set of element orders is said to solved for a group $G$ if we know the number of different finite groups having the same set of element orders as $G$. The goal of the talk is to survey recent results concerning this recognition problem, with particular emphasis given to recognition of finite simple groups. Also we discuss the related problem of finding the element orders of finite almost simple groups.