Electrons carry both charge and spin. The processing of information in conventional electronic devices is based only on the charge of the electrons. Spin electronics, or spintronics, uses the spin of electrons, as well as their charge, to process information. Metals, semiconductors and insulators are the basic materials that constitute the components of electronic devices, and these have been transforming all aspects of society for over a century. In contrast, magnetic metals, half-metals, magnetic semiconductors, dilute magnetic semiconductors and magnetic insulators are the materials that will form the basis for spintronic devices. Materials having a zero-energy band gap are a special class of these materials that exhibit some fascinating and superior electronic properties compared to materials with a non-zero energy gap. This article reviews a range of materials with zero-gap band structures, focusing on materials with quadratic and linear symmetrical dispersions, disorder-induced linear dispersions, asymmetrical linear dispersions and topological insulating states. These materials all have intriguing physical properties and numerous potential practical applications in spintronics, electronics, optics and sensors.