We review our current understanding of nuclear stellar discs (NSDs), rotating, and flattened stellar structures found in the central regions of both early- and late-type galaxies. We examine their demographics, kinematics, stellar populations, metallicity gradients and star formation histories. We derive scaling relations linking NSDs to properties of their host galaxies, and compare them with analogous relations for nuclear star clusters. The relationship between NSDs and other central galactic components, including nuclear rings, nuclear bars, and nuclear star clusters, is explored. The role of NSDs as tracers of the secular evolution of barred galaxies is highlighted, emphasising how they can be used to constrain properties of galactic bars such as their ages. Special attention is given to the Milky Way's NSD, which serves as a unique case study thanks to its proximity and the ability to resolve individual stars. The review covers the Milky Way's NSD structure, kinematics, dynamics and stellar content, addressing ongoing debates about the presence of a nuclear bar and implications of the latter for central gas dynamics. We argue that NSDs form by in-situ star formation, most of them because of bar-driven gas inflow, but possibly in some cases because of external acquisition of gas during a gas-rich merger. The review concludes by outlining open questions, future research directions and the exciting prospects provided by upcoming observational facilities.