Lithium-ion batteries are all around us. They are used for energy storage in portable electronics, hybrid-electric vehicles, even the Curiosity Mars rover. As we reduce our dependence on fossil fuels, the use of lithium-ion batteries will only increase. Today's commercial lithium-ion batteries, however, face problems of short lifespans - the degrading charge-time of a laptop or phone battery - and safety issues - illustrated recently by the fires reported on aircraft and in electric cars. If lithium-ion batteries are to be used more widely they must have both longer lifetimes and improved s!afety characteristics. One technology that may potentially address is an "all-solid-state" lithium-ion battery. Conventional batteries have liquid electrolytes that transport lithium ions between the cathode and anode as the battery is charged or discharged. These electrolytes are chemically unstable, and degrade with time - leading to reduced capacity - or can react explosively in the case of a short-circuit or intense heating. In an all-solid-state battery, these are replaced by ceramic solids that are chemically inert and robust, yet still allow lithium ions to move between the electrodes. The challenge is to develop solid electrolytes that meet these specifications of chemical stability and good lithium-ion conductivity, while also being electronically insulating. One of the limiting factors for battery performance is the rate at which lithium can move through the device: this dictates the speed with which it charges and the total power density of the battery. a specific issue for all-solid-state batteries is that even in the case of solid electrolytes that allow rapid lithium transport, transferring lithium between the electrolyte and an electrode (or vice-versa) can be slow, ultimately limiting the overall lithium transport rate. This ability of the boundary between electrode and electrolyte to impede the conduction of lithium ions is the "interfacial resistance", which should be low for high-power applications. In 1985 a patent was filed that proposed a strategy for designing solid-state batteries with low interfacial resistance: if the electrodes and electrolyte all have structures based on the same underlying crystal lattice, then it should be possible to find combinations of materials that are "lattice-matched", with similar crystal lattice dimensions for the electrodes and electrolyte. It was proposed that this would allow the lithium- conduction pathways to line up across the electrode-electrolyte interfaces, forming continuous channels with low interfacial resistance, and enabling fast ionic transport through the battery. The application of this idea has until recently been limited by an absence of promising solid electrolytes with crystal structures that permit lattice-matching with known solid electrodes. In 2013, however, a new family of solid-electrolytes was reported with the same underlying "spinel-structure" as a number of widely studied solid electrodes, breathing new life into this concept. This research project seeks to use computer simulations to understand how the chemical composition of spinel-structured lithium-electrolytes affects their suitability for lattice-matched solid-state batteries. We will construct new models that are able to describe numerically the motions and interactions of atoms that make up these materials. These models will then be used in large-scale computer simulations to study how differences in chemistry control the details of crystal structure and rates of lithium ion conduction in these materials. By extending these models to also describe electrode-electrolyte interfaces, we will directly calculate interfacial resistances, and explore the potential for "lattice-matching" as a strategy for the design of high-performance solid-state lithium-ion batteries.