This work studies multi-objective optimization (MOP) of nonsmooth functions subject to general constraints. We first present definitions and optimality conditions as well as some single-objective formulations of MOP, parameterized with respect to some reference point in the space of objective functions. Next, we propose a new algorithm called MultiMads (multi-objective mesh adaptive direct search) for MOP. MultiMads generates an approximation of the Pareto front by solving a series of single-objective formulations of MOP generated using the NBI (natural boundary intersection) framework. These single-objective problems are solved using the Mads (mesh adaptive direct search) algorithm for constrained nonsmooth optimization. The Pareto front approximation is shown to satisfy some first order necessary optimality conditions based on the Clarke calculus. MultiMads is then tested on problems from the literature with different Pareto front landscapes and on a styrene production process simulation problem from chemical engineering.