  Axial algebras are a class of non-associative commutative algebras whose
properties are defined in terms of a fusion law. When this fusion law is
graded, the algebra has a naturally associated group of automorphisms and thus
axial algebras are inherently related to group theory. Examples include most
Jordan algebras and the Griess algebra for the Monster sporadic simple group.
  In this survey, we introduce axial algebras, discuss their structural
properties and then concentrate on two specific classes: algebras of Jordan and
Monster type, which are rich in examples related to simple groups.
