Over the past fifteen years or so, the study of f-element single-ion magnets (f-SIMs) has gone from being a sub-discipline of molecular magnetism to an established field of research in its own right. The major driving force has been their exceptional promise in applications such as ultra-high-density data storage, spintronics, and quantum information processing (QIP). Recent demonstrations that f-SIMs preserve their intrinsic magnetic properties even when deposited onto substrates have reinforced the interests in the field.Here, we review the current state of the field of lanthanide and actinide f-SIMs; discuss the principal factors affecting the magnetic and quantum properties of such single-ion magnets; review the latest chemical approaches in designing f-SIMs with superior properties; and highlight new trends in single molecule magnetism, including using f-SIMs as potential spin qubits for quantum computers.