Although layered lithium transition-metal oxides with different compositions of (Ni,Co,Mn) [NCM] or (Ni,Co,Al) [NCA] have been used in commercial lithium-ion battery cathodes, their defect physics and chemistry is still not well understood, despite having important ramifications for cycling properties, particularly capacity fade. Herein we report a hybrid density functional study of the crystal and electronic structures of and intrinsic point defects in the compositions LiNi1/3Co1/3Mn1/3O2 (NCM1/3) and LiNi1/3Co1/3Al1/3O2 (NCA(1/3)), which also serve as model compounds for NCM and NCA. We find that the transition metals can exist in different charge and spin states at different lattice sites. In NCM1/3, nickel/lithium antisite pairs, i.e., Ni-Li-Li-N, are estimated to be about 3% in samples prepared at 1000 degrees C. In NCA(1/3), in addition to nickel antisites Ni-Li, aluminum antisites Al-Li can also occur with a very high concentration. The Al-Li defect has a high energy except when it is located between two Al atoms at the transition-metal sites both above and below the lithium layer, i.e., when the concentration of Al is high enough that significant amounts exist in three successive layers. Our results thus provide a natural explanation for why the observed improvement in the electrochemical performance of NCA at low Al concentrations gives way to drastically decreased performance beyond about 10%. The effects of substituting Al for Mn on the lithium migration and overall voltage profile are also discussed.