This chapter introduces the nonlinear normal modes (NNMs) for vibrating systems as invariant manifolds of the phase space, and their use for model order reduction of nonlinear structures. NNMs are defined as the continuation of the linear normal modes by enforcing tangency to a subset of master eigenspaces for small amplitudes. Conservative and damped dynamics are considered, as well as forced systems where the NNMs are time-dependent. A systematic procedure using the parametrisation method for invariant manifolds, is devised for their computation, directly operating from the physical space, and up to arbitrary order of expansions. Applications to academic examples are shown to highlight the ability of the method to deal with hardening/softening behaviour, the presence of a folding manifold, and superharmonic resonance. In each case, reduced-order models with minimal dimensions and excellent accuracy, are derived.