Let X be a complex algebraic variety, and X à D a proper morphism to a small disk which is smooth away from the origin. In this setting, the higher direct images of the constant sheaf form a local system on the punctured disk, and the Local Monodromy Theorem (due to Brieskorn-Grothendieck-Griffiths-Landsman) asserts that the eigenvalues of local monodromy are roots of unity. In this talk, we will discuss generalizations of this result to the setting of arbitrary morphisms between complex algebraic varieties, and with coefficients in arbitrary constructible sheaves. If there is time, I'll discuss applications to variation of monodromy in abelian covers, and applications to the monodromy of alexander modules. This is based on joint work with Madhav Nori.