International audience ; We consider optimization problems with objective and constraint being the difference of convexand weakly convex functions. This framework covers a vast family of nonsmooth and nonconvex optimizationproblems, particularly those involving certain classes of composite and nonconvex value functions. Weinvestigate several stationary conditions and extend the proximal bundle algorithm of [van Ackooij et al.,Comput. Optim. Appl., 78 (2021), pp. 451–490 ] to compute critical points for problems in this class. Ourmodifications on that algorithm boil down to a different master program and an original rule to updatethe proximal parameter to ensure convergence in this more general setting. Thanks to this new rule, nopre-estimation of the underlying weakly-convex moduli is needed, opening the way to deal with optimizationproblems for which no practical and mathematically sound algorithms exist. Numerical experiments on somenonconvex stochastic problems illustrate the practical performance of the method.