Fock and Goncharov showed that the moduli space of decorated SL(k)-local systems on a bordered marked surface is a cluster variety. When k is 2, the resulting cluster algebras are the cluster algebras from surfaces studied by Fomin, Shapiro, and Thurston. In this case, clusters are in bijection with tagged triangulations of the surface. We propose a conjectural way of extending these combinatorial structures to higher k, and present results in support of our conjectures. The main inputs to our work are due to Goncharov-Shen and Fomin-Pylyavskyy. This is joint work with Pavlo Pylyavskyy. Speaker(s): Christopher Fraser (Michigan State)