Expansive systems are a class of topological dynamical systems which exhibit sensitivity to initial conditions. Following work by Klaus Thomsen, we will construct a C*-algebra from an asymptotic equivalence relation on the points in a given expansive system. This generalizes the C*-algebraic construction made by Ian Putnam for Smale Spaces, which are themselves an important special case of expansive systems. We will then present some results obtained for a type of expansive system called synchronizing systems, and we will compute this C*-algebra for some specific examples. No knowledge of dynamical systems will be assumed for this talk.