The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial f over the integers, more precisely, poles of the p-adic Igusa zeta function of f should induce monodromy eigenvalues of f. The case of interest is when the zero set of f has singular points. We first present some history and motivation. Then we expose a proof in the case of two variables, and partial results in higher dimension, together with geometric theorems of independent interest inspired by the conjecture. We conclude with several possible generalizations.