We present the BACCO project, a simulation framework specially designed to provide highly-accurate predictions for the distribution of mass, galaxies, and gas as a function of cosmological parameters. In this paper, we describe our main suite of simulations (L $sim2$ Gpc and $4320^3$ particles) and present various validation tests. Using a cosmology-rescaling technique, we predict the nonlinear mass power spectrum over the redshift range $0<z<1.5$ and over scales $10^{-2} < k/(h Mpc^{-1} ) < 5$ for 800 points in an 8-dimensional cosmological parameter space. For an efficient interpolation of the results, we build an emulator and compare its predictions against several widely-used methods. Over the whole range of scales considered, we expect our predictions to be accurate at the 2% level for parameters in the minimal $Lambda$ CDM model and to 3% when extended to dynamical dark energy and massive neutrinos. We make our emulator publicly available under http://www.dipc.org/bacco
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