In this paper we study the semileptonic decays of B + c → D(s) (l + ν l , l + l − , νν) (here l stands for e, µ, or τ ). After evaluating the B + c → (D (s) , D * (s) ) transition form factors F 0,+,T (q 2 ) and V (q 2 ), A 0,1,2 (q 2 ), T 1,2,3 (q 2 ) by employing the perturbative QCD factorization approach, we calculate the branching ratios for all these semileptonic decays. Our predictions for the values of the B + c → D (s) and B + c → D * (s) transition form factors are consistent with those obtained by using other methods. The branching ratios of the decay modes withνν are almost an order of magnitude larger than the corresponding decays with l + l − after the summation over the three neutrino generations. The branching ratios for the decays with b → d transitions are much smaller than those decays with the b → s transitions, due to the Cabibbo-Kobayashi-Maskawa suppression. We define ratios R D and R D * for the branching ratios with the τ lepton versus µ, e lepton final states to cancel the uncertainties of the form factors, which could possibly be tested in the near future.