Abstract: If you randomly pick an automorphism of a mathematical object $M$ (a bijection from $M$ to $M$ that preserves whatever structure $M$ is endowed with), what will likely be true of its centralizer, i.e. the set of automorphisms that commute with it? We will discuss this question, and its answer, for several objects $M$. Emphasis will be given to the group $Aut(X,\mu$) of invertible measure preserving transformations of a non-atomic probability space $(X,\mu)$, where the central question remains open: Does there exist a topological group $G$ so that for almost every transformation $T$, the centralizer of $T$ is isomorphic to $G$?