"Free Discontinuity Problems - Mumford Shah type Problems" (Solvability, Partial Regularity and Monotonicity Formulae) Abstract: We extend and simplify the result by Carriero, De Giorgi, Leaci that (almost) minimizers of the (relaxed) Mumford Shah functional are regular outside a closed one codimensional set. The connection with the theory of minimal surfaces - GMT - and the method of monotonicity formulae will be explained. (Joint work with Dorin Bucur)