We will describe how Laplacians, Dirichlet forms and continuous diffusion processes can be constructed on many self-similar finitely ramified fractals. If the fractal have plenty of symmetries, then the spectrum and even eigenfunctions can be computed explicitly. One of the applications is the possibility to do analysis on limit sets of self-similar groups. If time permits, some of the unresolved questions will be mentioned, such as the difficulties in defining the first order derivatives.