Let M=M(g) be the mapping class group of a surface of genus g > 1 (resp. M=Aut(F_g) the automorphism group of the Free group on g generators ). As it is well known, M is mapped onto the symplectic group Sp(2g,Z) (resp. the general linear group GL(g,Z) ). We will show that this is only a first case in a series: in fact, for every pair (S,r) when S is a finite group with less than g generators and r is a Q-irreducible representation of S, we associate an arithmetic group which is then shown to be a virtual quotient of M. The case when S is the trivial group gives the above Sp(2g,Z) ( resp. GL(g,Z) ) but many new quotients are obtained. For example it is used to show that M(2) (resp. Aut(F_3) ) is virtually mapped onto a non-abelian free group. Another application is an answer to a question of Kowalski: generic elements in the Torelli groups are hyperbolic and fully irreducible. Joint work with Fritz Gruenwald, Michael Larsen and Justin Malestein .