We present an investigation of the phase stability, electrochemical stability and Li+ conductivity of the Li10±1MP2X12 (M = Ge, Si, Sn, Al or P, and X = O, S or Se) family of superionic conductors using first principles calculations. The Li10GeP2S12 (LGPS) superionic conductor has the highest Li+ conductivity reported to date, with excellent electrochemical performance demonstrated in a Li-ion rechargeable battery. Our results show that isovalent cation substitutions of Ge4+ have a small effect on the relevant intrinsic properties, with Li10SiP2S12 and Li10SnP2S12 having similar phase stability, electrochemical stability and Li+ conductivity as LGPS. Aliovalent cation substitutions (M = Al or P) with compensating changes in the Li+ concentration also have a small effect on the Li+ conductivity in this structure. Anion substitutions, however, have a much larger effect on these properties. The oxygen-substituted Li10MP2O12 compounds are predicted not to be stable (with equilibrium decomposition energies >90 meV per atom) and have much lower Li+ conductivities than their sulfide counterparts. The selenium-substituted Li10MP2Se12 compounds, on the other hand, show a marginal improvement in conductivity, but at the expense of reduced electrochemical stability. We also studied the effect of lattice parameter changes on the Li+ conductivity and found the same asymmetry in behavior between increases and decreases in the lattice parameters, i.e., decreases in the lattice parameters lower the Li+ conductivity significantly, while increases in the lattice parameters increase the Li+ conductivity only marginally. Based on these results, we conclude that the size of the S2− is near optimal for Li+ conduction in this structural framework.