The architecture and operation of the human brain serve as the inspiration for the computer technique known as neuromorphic computing. Any device that does computations using real, artificial neurons is called a neuromorphic computer or chip. The final major frontier of computing is the brain. By drawing inspiration from the brain, neuromorphic computing produces extraordinarily complex, energy-efficient hardware and information processing. This chapter reviews some of the most recent advances in new memory technologies and explains how they could be applied to create neuromorphic computing systems that are extremely energy-efficient. Compared to most super computers, which occupy entire rooms, the human brain is small and readily fits within something that is, well, the size of your skull. This makes it an especially interesting in line for computing.