We compare the physical properties and solution conductivities of three new lithium orthoborate salts with those of the well-known salt lithium bis(trifluoromethanesulfonyl)imide (LiTFSI). The three new lithium salts are lithium bis(perfluoropinacolato)borate (LiBPFPB), lithium bis(oxalato)borate, and lithium bis(malonato)borate (LiBMB). Computational models of the three orthoborate anions show that the borate oxygens in BPFPB- anion are the least exposed. The oxygens are electronically identical in BPFPB- but not in the other anions. The three new lithium salts show conductivities that closely approach those of LiTFSI but show surprising and solvent-dependent orderings. The conductivity is nearly independent of the salt content in the salt concentration range of 0.5-1 M, which is advantageous for their applications. Self-diffusivity measurements for Li-7, F-19, and H-1 are presented and are consistent with the very high ionic dissociation levels proposed for these salts based on other evidence. The lithium-ion transport number at room temperature for LiBPFPB in nonaqueous solutions exceeds 0.5 and for LiBMB is about 0.4.