In this talk, I will discuss cluster algebras from surfaces, as defined by Fomin-Shapiro-Thurston, based on earlier work of Gekhtman-Shapiro-Vainshtein and Fock-Goncharov. Generators for such algebras correspond to arcs on a Riemann surface (possibly with boundary), and can be expressed as Laurent polynomials. As shown by Fomin-Thurston, the cluster variables also correspond to lambda lengths, as in Penner's description of decorated Teichmuller space.  In joint work with Schiffler and Williams, we constructed vector-space bases for many such cluster algebras, giving combinatorial formuals (with positive coefficients) for these Laurent expansions in the process.