  Let $(M,g)$ be a closed Riemannian manifold and $L:TM\rightarrow \mathbb R$
be a Tonelli Lagrangian. In this thesis we study the existence of orbits of the
Euler-Lagrange flow associated with $L$ satisfying suitable boundary
conditions. We first look for orbits connecting two given closed submanifolds
of $M$ satisfying the conormal boundary conditions: We introduce the Ma\~n\'e
critical value that is relevant for the problem and prove existence results for
supercritical and subcritical energies; we also complement these with
counterexamples, thus showing the sharpness of our results. We then move to the
problem of finding periodic orbits: We provide an existence result of periodic
orbits for non-aspherical manifolds generalizing the Lusternik-Fet Theorem, and
a multiplicity result in case the configuration space is the 2-torus.
