  Elasticity typically refers to a material's ability to store energy, while
viscosity refers to a material's tendency to dissipate it. In this review, we
discuss fluids and solids for which this is not the case. These materials
display additional linear response coefficients known as odd viscosity and odd
elasticity. We first introduce odd elasticity and odd viscosity from a
continuum perspective, with an emphasis on their rich phenomenology, including
transverse responses, modified dislocation dynamics, and topological waves. We
then provide an overview of systems that display odd viscosity and odd
elasticity. These systems range from quantum fluids, to astrophysical gasses,
to active and driven matter. Finally, we comment on microscopic mechanisms by
which odd elasticity and odd viscosity arise.
