Non-linear normal modes (NNMs), defined as invariant manifolds, are introduced through Normal Form theory. In a conservative framework, it is shown that all NNMs, as well as the attendant dynamics onto the manifolds, are computed in a single operation. The general third-order approximation of the dynamics onto a single NNM is derived. It is underlined that single linear mode truncation can lead to erroneous results which are corrected when considering NNMs. These results are illustrated by studying the vibrations of a linear beam resting on a non-linear elastic foundation.