This work aims to solve numerically non-linear partial differential equations on surfaces, that may evolve in time, for a set of different applications. The core of all the numerical schemes is a finite element method recently introduced for triangulated surfaces. The main classes of applications under appreciation are the motion of curves on surfaces, segmentation of images painted on surfaces and the formation of Turing patterns on surfaces. For the first one, three different approaches are considered and compared: the level set method, the phase field framework and the diffusion generated motion method. The formation of patterns leads to an interesting application to the growth of tumours which is also investigated. Implementation of all numerical schemes proposed is carried out and some analysis on the convergence and stability of the approximations is presented. The finite element method has shown efficiency and great flexibility when it comes to the equations it can approximate and the surfaces it can handle.