This paper discusses fast and accurate methods to solve total variation (TV) models on the graphics processing unit (GPU). We review two prominent models incorporating TV regularization and present different algorithms to solve these models. We mainly concentrate on variational techniques, i.e. algorithms which aim at solving the Euler Lagrange equations associated with the variational model. We then show that particularly these algorithms can be effectively accelerated by implementing them on parallel architectures such as GPUs. For comparison we chose a state-of-the-art method based on discrete optimization techniques. We then present the results of a rigorous performance evaluation including 2D and 3D problems. As a main result we show that the our GPU based algorithms clearly outperform discrete optimization techniques in both speed and maximum problem size.