  A fundamental property of compact groups and compact quantum groups is the
existence and uniqueness of a left and right invariant probability -- the Haar
measure. This is a natural playground for classical and quantum probability,
provided it is possible to compute its moments. Weingarten calculus addresses
this question in a systematic way. The purpose of this manuscript is to survey
recent developments, describe some salient theoretical properties of Weingarten
functions, as well as applications of this calculus to random matrix theory,
quantum probability, and algebra, mathematical physics and operator algebras.
