  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.
