  The hybrid high-order method is a modern numerical framework for the
approximation of elliptic PDEs. We present here an extension of the hybrid
high-order method to meshes possessing curved edges/faces. Such an extension
allows us to enforce boundary conditions exactly on curved domains, and capture
curved geometries that appear internally in the domain e.g. discontinuities in
a diffusion coefficient. The method makes use of non-polynomial functions on
the curved faces and does not require any mappings between reference
elements/faces. Such an approach does not require the faces to be polynomial,
and has a strict upper bound on the number of degrees of freedom on a curved
face for a given polynomial degree. Moreover, this approach of enriching the
space of unknowns on the curved faces with non-polynomial functions should
extend naturally to other polytopal methods. We show the method to be stable
and consistent on curved meshes and derive optimal error estimates in $L^2$ and
energy norms. We present numerical examples of the method on a domain with
curved boundary, and for a diffusion problem such that the diffusion tensor is
discontinuous along a curved arc.
