In this chapter, we explain the basic ideas and principles of nuclear spins and resonance, NMR quantum gates and operations. Liquid NMR quantum computing may be regarded as the progenitor of the general quantum computing technology. Its techniques have deeply influenced the development of today’s QC and, thus, in chronological order, we should have placed this chapter in front of most of the chapters on QC devices. However, the liquid NMR technology is faced with the very uncertain future of getting phased out (due to its lack of scalability). Presently, a revival of NMR quantum computing is taking place, making new progress in optically addressed solid state NMR, which is expounded. Examples of the Shor’s algorithm for factorization of composite integers and quantum lattice-gas algorithm for the diffusion partial differential equation are also illustrated.