The purpose of this thesis is to study the spectrum of sublaplacians on compact strictly pseudoconvex CR manifolds. We prove the discreteness of the Dirichlet spectrum of the sublaplacian $\Delta_b$ on a smoothly bounded domain $\Omega \subset M$ in a strictly pseudoconvex CR manifold M satisfying Poincaré inequality. We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set ${\mathcal P}+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}+$ with a natural metric topology. We establish inequalities for the eigenvalues of $\Delta_b$ on compact strictly pseudoconvex CR manifolds (possibly with nonempty boundary) %$C^2$ semi-isometric maps into a Euclidean space or a Heisenberg group. Our estimates extend those obtained by P-C. Niu \& H. Zhang \cite{NiZh} for the Dirichlet eigenvalues of the sublaplacian on a bounded domain in the Heisenberg group, in the spirit of Payne-P\'{o}lya -Weinberger and Yang inequalities. We establish a new lower bound on the first nonzero eigenvalue $\lambda_1 (\theta )$ of the sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$ carrying a contact form $\theta$ whose Tanaka-Webster connection has Ricci curvature bounded from below.
