This survey presents some applications of KAM theory and variational methods to some concrete problems arising in celestial mechanics and PDEs. While the details of the proofs, sometimes lengthy or technical, are published elsewhere, we provide here a general overlook of results and techniques, with the aim of accenting the analogies between the finite and infinite dimensional cases and the interplay of the KAM-Nash-Moser iteration techniques with the Lyapunov-Schmidt variational methods. In this review, we deal with four topics: Spatial planetary three-body problem. We consider one “star” and two “planets”, modelled by three massive points, interacting through gravity in a three-dimensional space. Near the limiting solutions given by the two planets revolving around the star on Keplerian ellipses with small eccentricity and small non-zero mutual inclination, the system is proved to have two-dimensional, elliptic, quasiperiodic solutions, provided the masses of the planets are small enough compared to the mass of the star and provided the osculating Keplerian major semiaxes belong to a two-dimensional set of density close to one. Planar planetary many-body problem. As above, but one “star” and N “planets”, the interior two ones bigger than the others (as in the exterior solar system). Near the limiting solutions given by the N planets revolving around the star on Keplerian ellipses with small eccentricity and zero mutual inclination, the system is proved to have N -dimensional, elliptic, quasiperiodic solutions. Periodic orbits approaching lower dimensional elliptic KAM tori. By a general Birkhoff-Lewis-Conley-Zehnder-type result, we prove the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori of Hamiltonian systems. As an application, periodic orbits close to the quasiperiodic ones of the above planetary problems are constructed.