  This paper addresses the factorization problem of almost simple groups,
studies regular subgroups of quasiprimitive permutation groups, and
characterizes $s$-arc transitive Cayley graphs of finite simple groups.
  The main result of the paper presents a classification of exact
factorizations of almost simple groups, which has been a long-standing open
problem initiated around 1980 by the work of Wiegold-Williamson (1980), and
significantly progressed by Liebeck, Praeger and Saxl in 2010.
  A by-product theorem of classifying exact factorizations of almost simple
groups shows that, if $G=HK$ is a classical group of Lie type and $H,K$ are
nonsolvable, then $H$ or $K$ is `almost maximal', with only a couple of
exceptions.
  As applications of the classification result,
  (1) a classification is given of quasiprimitive groups containing a regular
simple group;
  (2) a characterization is obtained for $s$-arc transitive Cayley graphs of
simple groups, showing that $3$-arc transitive Cayley graphs of simple groups
are surprisingly rare;
  (3) a question regarding biperfect Hopf algebras is confirmed.
