Connecting orbits describe the paths along which changes occur in deterministic systems. In fact, every such dynamical system can be decomposed into so-called recurrent components and connecting orbits between them. This is most pronounced in gradient systems, where this dichotomy between equilibria and connections leads to the definition of the Morse-Floer complex. While equilibria are relatively manageable objects analytically, one usually has to resort to numerical simulations for studying connecting orbits. And while numerical calculations offer clear pictures of the dynamics, the information the numerics provide is non-rigorous. Strikingly, today's advances in computer speed and algorithm development make it possible to utilize the power and robustness of topological-analytic methods to rigorously validate computational results. Indeed, the past decade has seen enormous advances in the development of computer assisted proofs in dynamics. In this talks we will discuss recent progress in the rigorous computation of connecting orbits, and directions for further research will be outlined.