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The conjugacy relation plays an important role in group theory. If $a$ and$b$ are elements of a group~$G$, $a$ is conjugate to $b$ if $g^{-1}ag=b$ forsome $g\in G$. Group conjugacy extends to inverse semigroups in a natural way:for $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$ if$g^{-1}ag=b$ and $gbg^{-1}=a$ for some $g\in S$. The fourth author has recentlydefined a conjugacy for an arbitrary semigroup $S$ that coincides with inversesemigroup conjugacy if $S$ is an inverse semigroup, and is included in allexisting semigroup conjugacy relations. We will call it the \emph{naturalconjugacy} for semigroups, and denote it by $\cfn$.  The first purpose of this paper is to study $\cfn$ in various contexts,chiefly the partition monoid and some of its friends (Brauer and partial Brauermonoids), and also to characterize $\cfn$ in several important classes ofsemigroups, transformation semigroups and in the polycyclic monoids.  The second purpose of this paper is to show how the notion of naturalconjugacy leads to the definition of the inverse semigroup of partialautomorphisms of an arbitrary semigroup (in the same way conjugation in groupsinduces the notion of inner automorphism). Attached to the majority ofmathematical objects there is a notion of {\em morphism} and hence notions ofautomorphism and endomorphism that often encode relevant information about theoriginal object. Our approach allows to attach to the endomorphisms of amathematical object an inverse semigroup that hopefully will bring the deepresults on inverse semigroups to help the study of the original object.  Finally we extend the notion of conjugacy growth from groups to semigroupsand give closed formulas for the conjugacy growth series of the polycyclicmonoid, for $\cfn$ and two other semigroup conjugacies.  The paper ends with some open problems.

Conjugacy in Semigroups: the Partition and Brauer Diagram Monoids, Conjugacy Growth, and Partial Inner Automorphisms