  This review is a primer on recently established geometric methods for
observables in quantum field theories. The main emphasis is on amplituhedra,
i.e. geometries encoding scattering amplitudes for a variety of theories. These
pertain to a broader family of geometries called positive geometries, whose
basics we review. We also describe other members of this family that are
associated with different physical quantities and briefly consider the most
recent developments related to positive geometries. Finally, we discuss the
main open problems in the field. This is a Topical Review invited by Journal of
Physics A: Mathematical and Theoretical.
