Forbidden optical transitions in single laser-cooled trapped ions make excellent references for accurate optical frequency standards, because the ion trap provides an environment in which external perturbations can be well controlled and characterised. This review provides an introduction to trapped-ion optical frequency standards, covering their principles of operation, the current state-of-the-art for the different standards being studied worldwide, and the various contributions to their systematic uncertainty budgets. Finally, future prospects for redefining the SI second in terms of an optical frequency standard are discussed.