The Cops and Robber game is a well-studied two-player pursuit-evasion game played on graphs, where a team of cops attempts to capture a robber. The cop number of a graph represents the minimum number of cops required for a successful capture. Graphs with a cop number of one are known as cop-win graphs. In this seminar, I will begin by characterizing cop-win graphs. Subsequently, I will demonstrate, using the technique of guarding a subgraph—a method for bounding the cop number of graphs with geometric representations—that the cop number for planar graphs is atmost three.