Neutrinos, being the only fermions in the Standard Model of Particle Physics that do not possess electromagnetic or color charges, have the unique opportunity to communicate with fermions outside the Standard Model through mass mixing. Such Standard Model-singlet fermions are generally referred to as "sterile neutrinos''. In this review article, we discuss the theoretical and experimental motivation for sterile neutrinos, as well as their phenomenological consequences. With the benefit of hindsight in 2020, we point out potentially viable and interesting ideas. We focus in particular on sterile neutrinos that are light enough to participate in neutrino oscillations, but we also comment on the benefits of introducing heavier sterile states. We discuss the phenomenology of eV-scale sterile neutrinos in terrestrial experiments and in cosmology, we survey the global data, and we highlight various intriguing anomalies. We also expose the severe tension that exists between different data sets and prevents a consistent interpretation of the global data in at least the simplest sterile neutrino models. We discuss non-minimal scenarios that may alleviate some of this tension. We briefly review the status of keV-scale sterile neutrinos as dark matter and the possibility of explaining the matter-antimatter asymmetry of the Universe through leptogenesis driven by yet heavier sterile neutrinos.

Submitted as an invited review for Physics Reports