We present the largest set of N-body and hydrodynamical simulations to date for cosmological models featuring a direct interaction between the dark energy (DE) scalar field, responsible for the observed cosmic acceleration, and the cold dark matter (CDM) fluid. With respect to previous works, our simulations considerably extend the statistical significance of the simulated volume and cover a wider range of different realizations of the interacting DE scenario, including the recently proposed bouncing coupled DE model. Furthermore, all the simulations are normalized in order to be consistent with the present bounds on the amplitude of density perturbations at last scattering, thereby providing the first realistic determination of the effects of a DE coupling for cosmological growth histories fully compatible with the latest cosmic microwave background data. As a first basic analysis, we have studied the impact of the coupling on the non-linear matter power spectrum and on the bias between the CDM and baryon distributions, as a function of redshift and scale. For the former, we have addressed the issue of the degeneracy between the effects of the coupling and other standard cosmological parameters, e.g. σ8, showing how the redshift evolution of the linear amplitude or the scale dependence of the non-linear power spectrum might provide a way to break the degeneracy. For the latter, instead, we have computed the redshift and scale dependence of the bias in all our different models showing how a growing coupling or a bouncing coupled DE scenario provides much stronger effects with respect to constant coupling models. Furthermore, we discuss the main features imprinted by the DE interactions on the halo and subhalo mass functions. We refer to this vast numerical initiative as the COupled Dark Energy Cosmological Simulations (codecs) project, and release all the codecs outputs for public use through a dedicated web data base, providing information on how to access and interpret the data.