Wall Cops and Robbers is a new turn-based game played on graphs. It is inspired by the games of Cops and Robbers and the Angel Problem. The objective of the game is for the cops to capture the robber by surrounding him with walls. The wall cop number of a graph G, written Wc(G), is the least number of cops it takes to capture the robber in G. The thesis explores these two parameters for various graph classes including trees, hypercubes, grids, and tilings of the plane. We also introduce a variant of the game called Wall Cops and Wall Robbers. In this game, the robber creates a wall on visiting a vertex, disallowing a move to a previously visited vertex.