Recent research has shown that certain Li-oxide garnets with high mechanical, thermal, chemical, and electrochemical stability are excellent fast Li-ion conductors. However, the detailed crystal chemistry of Li-oxide garnets is not well understood, nor is the relationship between crystal chemistry and conduction behavior. An investigation was undertaken to understand the crystal chemical and structural properties, as well as the stability relations, of Li7La3Zr2O12 garnet, which is the best conducting Li-oxide garnet discovered to date. Two different sintering methods produced Li-oxide garnet but with slightly different compositions and different grain sizes. The first sintering method, involving ceramic crucibles in initial synthesis steps and later sealed Pt capsules, produced single crystals up to roughly 100 mu m in size. Electron microprobe and laser ablation inductively coupled plasma mass spectrometry (ICP-MS) measurements show small amounts of Al in the garnet, probably originating from the crucibles. The crystal structure of this phase was determined using X-ray single-crystal diffraction every 100 K from 100 K up to 500 K. The crystals are cubic with space group la $(3) over bar $d at all temperatures. The atomic displacement parameters and Li-site occupancies were measured. Li atoms could be located on at least two structural sites that are partially occupied, while other Li atoms in the structure appear to be delocalized. Al-27 NMR spectra show two main resonances that are interpreted as indicating that minor Al occurs on the two different Li sites. Li NMR spectra show a single narrow resonance at 1.2-1.3 ppm indicating fast Li-ion diffusion at room temperature. The chemical shift value indicates that the Li atoms spend most of their time at the tetrahedrally coordinated C (24d) site. The second synthesis method, using solely Pt crucibles during sintering, produced fine-grained Li7La3Zr2O12 crystals. This material was studied by X-ray powder diffraction at different temperatures between 25 and 200 degrees C. This phase is tetragonal at room temperature and undergoes a phase transition to a cubic phase between 100 and 150 degrees C. Cubic "Li7La3Zr2O12" may be stabilized at ambient conditions relative to its slightly less conducting tetragonal modification via small amounts of Al3+. Several crystal chemical properties appear to promote the high Li-ion conductivity in cubic Al-containing Li7La3Zr2O12. They are (i) isotropic three-dimensional Li-diffusion pathways, (ii) closely spaced Li sites and Li delocalization that allow for easy and fast Li diffusion, and (iii) low occupancies at the Li sites, which may also be enhanced by the heterovalent substitution Al3+ <-> 3Li.