A new branch-and-bound based algorithm for smooth nonconvex multiobjective optimization problems with convex constraints is presented. The algorithm computes an $(\varepsilon,\delta)$-approximation of all globally optimal solutions. We introduce the algorithm which uses selection rules, discarding and termination tests. The discarding tests are the most important aspect, as they examine in different ways whether a box can contain optimal solutions and determine by that the speed and effectiveness of the algorithm. We present a discarding test which combines techniques from the $\alpha$BB method from global scalar-valued optimization with outer approximation techniques from convex multiobjective optimization and the concept of local upper bounds from combinatorial multiobjective optimization. We apply the algorithm to several test instances as well as to an application in Lorentz force velocimetry.