Discrete linear and nonlinear inverse problems arise from many different imaging systems. These problems are ill-posed, which means, in most cases, that the solution is very sensitive to the data. Because the data usually contain errors produced by different imaging system parts (e.g., cameras, sensors, etc.), robust and reliable regularization methods must be developed for computing meaningful solutions. In some imaging systems, massive amounts of data are produced, making the data storage and computational cost of the inversion process intractable. In this talk, we will see different imaging systems, formulate the corresponding mathematical models, introduce regularization methods, and show theoretical and numerical results.