Abstract: We give a simple, efficient solution to the conjugacy problem in Thompson's groups F, T and V. Each conjugacy class corresponds to a certain trivalent, directed graph, which can also be drawn on a surface in the cases of F and T. This provides a unified approach to the conjugacy problem for these three groups. These graphs also give a description of the dynamics of the corresponding group elements acting as piecewise-linear functions. (this is joint work with James Belk from Texas A&M)