Properties of minimizers of the total variation and of the solutions of the total variation flow

We review some results obtained in collaboration with Vicent Caselles (formerly with DTIC, U. Pompeu Fabra, Barcelona) and Matteo Novaga (U. Pisa) and recent extensions obtained by the first author on the properties of the jump sets of solutions of total variation minimizing problems, including some results on explicit solutions in the 1D and radial cases. We will also describe jump inclusion properties for non-homogeneous perimeters (which appear in active contour energies) and discuss some open problems.