A general principle for dealing with the intrinsic numerical instability of most inverse problems is that of regularization. Among the regularization approaches the first one is Tikhonov regularization. Next, in 1992 Rudin, Osher & Fatemi introduced total variation regularization for the edge preserving denoising and deblurring of images. In the same years Perona & Malik introduced the scale-space concepts and the Partial Differential Equations (PDEs) approaches to image regularization. Since then, according to the features of the specific application problem, several functionals have been investigated in order to provide more reliable and efficient models. Regularization approaches for solving inverse and ill posed problems developed systematically, on the other hand in real applications many difficulties still arise, starting from the choice of suitable discretization schemes to the selection of the best solvers for the underlying computational kernels, until the final fast and robust implementation. In this talk we would like to point out some computational issues concerning the numerical solution of inverse problems with special emphasis on ongoing work on image restoration. Three approaches will be discussed: cascadic multilevel methods, alternating minimization methods and PDEs methods. Numerical results illustrate the different approaches in image denoising and deblurring.