Summary: A duality with zero gap for nonconvex optimization problems is presented. The first class of nonconvex problems, where local optima may not be global, are problems of quasi-convex minimization over a convex set. For this class a generalized Kuhn-Tucker condition is obtained, and the duality is similar to the Fenchel-Moreau-Rockafellar duality scheme. By duality, one can reduce the problem to solving a system of convex and quasiconvex inequalities. Unlike previous developments, these conjugation functionals and dual problems are defined on the dual space and involve no extra parameter. For more general nonconvex problems, such as a quasiconvex maximization over a compact set or a general minimization over the complement of a convex set, a duality with zero gap can be obtained as well. A zero gap in primal-dual pairs allows the development of primal-dual algorithms for nonconvex problems. The primal-dual algorithms are very suitable when the dual problem is simpler than the primal one.