In this work, we are studying the extended Torelli and Prym maps. The first part is on the fibers of the extended Torelli map. We give a description of the locus where the fibers have positive dimension. We also give an example answering Namikawas question about the locus of injectivity of the extended Torelli map. The second part studies the extended Prym map. First we translate the combi-natorial description of the indeterminacy locus given by Alexeev, Birkenhake and Hulek into a simple geometric criterion. Then we resolve the indeterminacy of the extended Prym map. We describe fibers of the toric resolution in several important cases, including when the dual graph is a double ring.