This talk is based on joint work with Luigi Lunardon. To every smooth and proper variety X with trivial canonical bundle over the field of complex Laurent series C((t)), one can attach its motivic zeta function, which measures how the variety degenerates as t goes to 0. I will show that this motivic zeta function is a birational invariant of X and deduce the birational invariance of the monodromy conjecture for X, the main open problem in this context, which predicts a relation between geometric and cohomological degeneration properties. The talk will include a gentle introduction to motivic zeta functions and an overview of known results.