A celebrated theorem of Serrin asserts that one overdetermined condition on the boundary is enough to obtain radial symmetry in the one-phase overdetermined torsion problem. It is also known that imposing just one overdetermined condition on the boundary is not enough to obtain radial symmetry in the corresponding multi-phase overdetermined problem. In this talk, we show that in order to obtain radial symmetry in the two-phase overdetermined torsion problem, two overdetermined conditions are needed. Finally, we show that this pattern does not extend to multi-phase problems with three or more layers, for which we show the existence of non-radial configurations satisfying infinitely many overdetermined conditions on the outer boundary.