We present a new grid of presupernova models of massive stars extending in mass between 13 and 120M_{sun}, covering four metallicities (i.e., [Fe/H]=0, -1, -2, and -3) and three initial rotation velocities (i.e., 0, 150, and 300km/s). The explosion has been simulated following three different assumptions in order to show how the yields depend on the remnant mass-initial mass relation. An extended network from H to Bi is fully coupled to the physical evolution of the models. The main results can be summarized as follows. (a) At solar metallicity, the maximum mass exploding as a red supergiant (RSG) is of the order of 17M{sun}_ in the nonrotating case, with the more massive stars exploding as Wolf-Rayet (WR) stars. All rotating models, conversely, explode as WR stars. (b) The interplay between the core He-burning and the H-burning shell, triggered by the rotation-induced instabilities, drives the synthesis of a large primary amount of all the products of CNO, not just ^14^N. A fraction of them greatly enriches the radiative part of the He core (and is responsible for the large production of F), and a fraction enters the convective core, leading therefore to an important primary neutron flux able to synthesize heavy nuclei up to Pb. (c) In our scenario, remnant masses of the order of those inferred from the first detections of gravitational waves (GW 150914, GW 151226, GW 170104, GW 170814) are predicted at all metallicities for none or moderate initial rotation velocities.