Within the framework of the complete theory of supersymmetry without R-parity, where all possible R-parity violating terms are admitted, we perform a systematic analytical study of all sources of neutrino masses up to ``direct one-loop" (defined explicitly below) level. In the passing, we present the full result for squark and slepton masses. In particular, there are interesting $LR$ squark and slepton mixings, which involve both bilinear and trilinear R-parity violating parameters. The existence and important phenomenological implications of such terms have been largely overlooked in previous studies. In particular, in the studies under which either one type of the couplings is assumed to vanish or neglected, the terms would not show up. The $LR$ mixings play a central role in neutrino mass generation. Our results look straight forward to be obtained, which, in our opinion, is an illustration of the effectiveness of our formulation adopted.