In the study of infinite dimensional dynamical systems exploring the dynamics in the entire phase space is impossible. One strategy to tackle this problem is to focus on a set of special solutions that act as organizing centers. To single out these solutions computer-assisted proofs are being developed to find, for example, fixed points, periodic orbits and connecting orbits between those. Computer-assisted proofs in dynamics combine the strength of scientific computing, nonlinear analysis, numerical analysis, applied topology, functional analysis and approximation theory. While in the past decade, these techniques have primarily been applied to ODEs, we are starting to witness their applicability for infinite dimensional nonlinear dynamics generated by partial differential equations (PDEs), integral equations, delay differential equations (DDEs), and infinite dimensional maps. In this talk we will present recent advances in this direction, with a special emphasize on the dynamics in the Navier-Stokes equations.