Total variation regularized data fitting problems arise from a number of image processing tasks, such as denoising, deconvolution, inpainting, magnetic resonance imaging, and compressive image sensing, etc. Recently, fast and efficient algorithms for solving such problems have been developing very rapidly. In this paper, we focus on least squares and least absolute deviation data fitting and present a brief algorithmic overview for these problems. We also discuss the application of a total variation regularized nonconvex data fitting problem in image restoration with impulsive noise.