Intercalation compounds form the basis of essentially all lithium rechargeable batteries. They exhibit a wide range of electronic and crystallographic structures. The former varies from metallic conductors to excellent insulators. The latter often have layer structures or have open tunnel structures that can act as the hosts for the intercalation of a wide range of metal cation and other guest species. They are fascinating materials with almost infinitely variable properties, with the crystal structure controlling the identity and the amount of the guest species that may be intercalated and subsequently removed. The electronic structure controls not only the degree of electron transfer to the host, but also defines the degree of the electrostatic interactions a mobile ion experiences; thus, a metallic host will provide a minimizing of those interactions, whereas in an ionic lattice the interactions will be much greater and the mobile ion will experience a much higher activation energy for motion. This becomes more important for multivalent cations such as Mg2+. Today's lithium batteries are limited in capacity, because less than one lithium ion is reversibly intercalated per transition metal redox center. There may be an opportunity to increase the storage capacity by utilizing redox centers that can undergo multielectron reactions. This might be accomplished by intercalating multiple monovalent cations or one multivalent cation. In this Account, we review the key theoretical and experimental results on lithium and magnesium reversible intercalation into two prototypical materials: titanium disulfide, TiS2, and vanadyl phosphate, VOPO4. Both of these materials exist in two or more phases, which have different molar volumes and/or dimensionalities and thus are expected to show a range of diffusion opportunities for battery active guest ions such as lithium, sodium, and magnesium. One major conclusion of this Account is that reversibly intercalating two lithium ions into a host lattice while maintaining its crystal structure is possible. A second major conclusion is that theoretical studies are now sufficiently mature that they can be relied upon to predict the key free energy values of simple intercalation reactions, i.e., the energy that might be stored. This could help to focus future choices of battery couples. In hindsight, theory would have predicted that magnesium-based intercalation cells are not a viable electrochemical option, relative to lithium cells, from either power or energy density considerations. However, the fundamental study of such reactions will lead to a better understanding of intercalation reactions in general, and of the critical importance of crystal structure in controlling the rates and degree of chemical reactions.