We consider the numerical solution of second order Partial Differential Equations (PDEs) on lower dimensional manifolds, specifically on surfaces in three dimensional spaces. For the spatial approximation, we consider Isogeometric Analysis which facilitates the encapsulation of the exact geometrical description of the manifold in the analysis when this is represented by B-splines or NURBS. Our analysis addresses linear, nonlinear, time dependent, and eigenvalues problems involving the Laplace–Beltrami operator on surfaces. Moreover, we propose a priori error estimates under h-refinement in the general case of second order PDEs on the lower dimensional manifolds. We highlight the accuracy and efficiency of Isogeometric Analysis with respect to the exactness of the geometrical representations of the surfaces.