Backlund transformations provide a method for contructing new solutions of a partial differential equation from a known solution. The new solutions are constructed by solving ordinary differential equations. These transformations are known to exist for certain special PDEs - in particular, for most integrable systems - but it is not known what conditions a PDE must satisfy in order to have a Backlund transformation. In this talk I will describe some classical Backlund transformations of hyperbolic Monge-Ampere equations in terms of exterior differential systems. Using Cartan's method of equivalence we can classify the homogenous examples, i.e., those transformations having maximal symmetry. In the process, we discover a previously unknown Backlund transformation between timelike surfaces of constant mean curvature in R{1,2}.