In this talk that is based on joint work with Luke Rogers and Robert Strichartz, I present a definition and and basic properties of pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to product of fractals. I will present some application to the study of elliptic, hypoelliptic, and quasi-elliptic operators, H\"ormander hypoelliptic operators as well as the study of wavefront sets and microlocal analysis on p.c.f. fractals.