This thesis uses the idea of lifting (or embedding) a nonlinear controlled dynamical system into an infinite-dimensional space of measures where this system is equivalently described by a linear equation. This equation and problems involving it are subsequently approximated using well-known moment-sum-of-squares hierarchies. First, we address the problems of region of attraction, reachable set and maximum controlled invariant set computation, where we provide a characterization of these sets as an infinite-dimensional linear program in the cone of nonnegative measures and we describe a hierarchy of finite-dimensional semidefinite-programming (SDP) hierarchies providing a converging sequence of outer approximations to these sets. Next, we treat the problem of optimal feedback controller design under state and input constraints. We provide a hierarchy of SDPs yielding an asymptotically optimal sequence of rational feedback controllers. In addition, we describe hierarchies of SDPs yielding approximations to the value function attained by any given rational controller, from below and from above, as well as a hierarchy of SDPs providing approximations from below to the optimal value function, hence obtaining performance certificates for the designed controllers as well as for any given rational controller. Finally, we describe a method to verify properties of a closed loop interconnection of a nonlinear dynamical system and an optimization-based controller (e.g., a model predictive controller) for deterministic and stochastic nonlinear dynamical systems. Properties such as global stability, the $\ell_2$ gain or performance with respect to a given infinite-horizon cost function can be certified. The methods presented are easy to implement using freely available software packages and are documented by a number of numerical examples.