A "central digraph" is a directed graph D such that given any two vertices x and y of D (not neccessarily distinct), there is a unique walk of length two from x to y. This occurs precisely when the adjacency matrix A of D has A2 = J (the matrix of all ones). These also correspond to an algebraic structure called a "central groupoid." In this talk we examine some of the basic properties of these objects as well as different methods of constructing them.