The reformulation of many of the classical constant-coefficient PDEs of mathematical physics (e.g. Laplace, Helmholtz, Maxwell, etc.) as boundary integral equations is a standard mathematical tool, which, when coupled with iterative solvers and fast algorithms such as fast multipole methods (FMMs), allows for the nearly optimal-time solution of these PDEs. Extending these methods to variable coefficient PDEs, especially those defined along surfaces, is not straightforward and ongoing research. In this talk, we will address the problem of solving the Laplace-Beltrami problem along surfaces in three dimensions. The Laplace-Beltrami problem is a variable coefficient PDE, with applications in electromagnetics, fluid-structure interactions, and surface diffusions. Our resulting integral equation is ready for acceleration using standard FMMs for Laplace potentials; several numerical examples will be provided.