  This chapter provides an introduction to Hybrid High-Order (HHO) methods.
These are new generation numerical methods for PDEs with several advantageous
features: the support of arbitrary approximation orders on general polyhedral
meshes, the reproduction at the discrete level of relevant continuous
properties, and a reduced computational cost thanks to static condensation and
compact stencil. After establishing the discrete setting, we introduce the
basics of HHO methods using as a model problem the Poisson equation. We
describe in detail the construction, and prove a priori convergence results for
various norms of the error as well as a posteriori estimates for the energy
norm. We then consider two applications: the discretization of the nonlinear
$p$-Laplace equation and of scalar diffusion-advection-reaction problems. The
former application is used to introduce compactness analysis techniques to
study the convergence to minimal regularity solution. The latter is used to
introduce the discretization of first-order operators and the weak enforcement
of boundary conditions. Numerical examples accompany the exposition.
