Polaritons are hybrid quasiparticles consisting of photons and electric (or magnetic) dipole-carrying excitations. Their observation has advanced our understanding of light–matter interactions and led to the manipulation of energy flows at the nanoscale. The discovery of strongly anisotropic van der Waals materials and the almost simultaneous development of near-field imaging techniques have led to the observation of a variety of highly confined polaritons with exotic properties, such as unidirectional, diffractionless or ray-like propagation, and hyperbolic dispersion. In this Review, we highlight the fundamental optical phenomena that have been redefined by these anisotropic polaritons, including anomalous cases of refraction, reflection and focusing. These phenomena promise unprecedented control over light–matter interactions at the nanoscale in spectral regions spanning from the visible to the terahertz. We also review strategies to manipulate these polaritons and offer our perspective on the challenges facing polaritonic research over the coming years towards practical applications.