[en] In this paper, we have studied a class of Hamiltonians that generalizes the Bose–Hubbard (BH) model of interacting bosons to include p-particle hopping as well as the multi-body interactions $U_{\mathrm{\ell <!--\; ???\; -->}}$ $(\mathrm{\ell <!--\; ???\; -->}⩾<!--\; ???\; -->2)$. While the case of two-body interaction $(U_2\equiv <!--\; ???\; -->U)$ and single-particle tunneling $(p=1)$ is well known, it is interesting to examine whether the resulting phase diagram for the general case has different features from that of the standard BH system. To this end we have applied a mean-field theory to locate critical lines, combined with the resolvent and Laplace transform method for the efficient calculation of the statistical sum for the quantum Hamiltonian. We have studied both ground state properties and temperature evolution of the phase diagram. Several new features, as an expansion of Mott insulator lobes for multi-body interaction and the emergence of a new insulating plateau for $p><!--\; >\; -->2$, have been demonstrated. These results may be of interest for experiments on cold atom systems in optical lattices. (paper)