  The action of any group on itself by conjugation and the corresponding
conjugacy relation play an important role in group theory. There have been many
attempts to find notions of conjugacy in semigroups that would be useful in
special classes of semigroups occurring in various areas of mathematics, such
as semigroups of matrices, operator and topological semigroups, free
semigroups, transition monoids for automata, semigroups given by presentations
with prescribed properties, monoids of graph endomorphisms, etc. In this paper
we study four notions of conjugacy for semigroups, their interconnections,
similarities and dissimilarities. They appeared originally in various different
settings (automata, representation theory, presentations or transformation
semigroups). Here we study them in maximum generality. The paper ends with a
large list of open problems.
