The monodromy conjecture of Denef—Loeser predicts that given a complex polynomial f, and any pole s of its motivic zeta function, exp(2πis) is a "monodromy eigenvalue" associated to f. In this talk I will formulate a geometric version of the conjecture​ and elaborate on ongoing work, starting from the case of Newton non-degenerate hypersurfaces. These are hypersurface singularities whose singularities are governed, up to a certain extent, by faces of their Newton polyhedra. The extent to which the former is governed by the latter is a key aspect of the conjecture. If time permits, I will also sketch a recent pursuit to reduce the conjecture to a setting that is slightly more general than the case of Newton non-degenerate hypersurfaces.