  The goal of this paper is to survey the history, development and current
status of the Return Times Theorem and its many extensions and variations. Let
$(X, \mathcal{F}, \mu)$ be a finite measure space and let $T:X \rightarrow X$
be a measure preserving transformation. Perhaps the oldest result in ergodic
theory is that of Poincar\'e's Recurrence Principle which states: For any set
$A \in \mathcal{F}$, the set of points $x$ of $A$ such that $T^nx$ is not in
the set $A$ for all $n > 0$ has zero measure. This says that almost every point
of $A$ returns to $A$. In fact, almost every point of $A$ returns to $A$
infinitely often.
  The return time for a given element $x \in A$, $r_A(x) = \inf\{k \geq 1: T^kx
\in A\}$, is the first time that the element $x$ returns to the set $A$. By
Poincar\'e's Recurrence Principle there is set of full measure in $A$ such that
all elements of this set have a finite return time. Our study of the Return
Times Theorem asks how we can further generalize this notion.
  The paper begins by looking at early work with the concept of weighted
averages. The second portion of the paper focuses on the historical development
of the proofs of the Return Times Theorem. Three areas of extension of the
Return Times Theorem are then considered: a multiterm version, characteristic
factors and breaking the H\"olderian duality. The paper concludes with
discussion of some more recent work and open questions to consider.
