  We introduce axial representations and modules over axial algebras as new
tools to study axial algebras. All known interesting examples of axial algebras
fall into this setting, in particular the Griess algebra whose automorphism
group is the Monster group. Our results become especially interesting for
Matsuo algebras. We vitalize the connection between Matsuo algebras and
3-transposition groups by relating modules over Matsuo algebras with
representations of 3-transposition groups. As a by-product, we define, given a
Fischer space, a group that can fulfill the role of a universal 3-transposition
group.
