The celebrated Monodromy Theorem states that the eigenvalues
of the monodromy of a polynomial are roots of unity. In this talk we
give on overview of recent results on local systems achieving a vast
generalization of the Monodromy Theorem. We end up with a conjecture
of Andre-Oort type for special loci of local systems. The conjecture
is true in rank one, and if true in general, it would provide a simple
proof in all generality of the DecompositionTheorem of
Beilinson-Bernstein-Deligne-Gabber. Joint work with Botong Wang.