The classical theory of partial differential equations (PDEs) on smooth, bounded domains--a great achievement of modern mathematics--is well understood and has many applications in both pure and applied mathematics. Many domains that arise in applications are, however, not smooth (that is, they do not have a smooth boundary). A rich and beautiful new theory of Analysis and Partial Differential Equations on Lipschitz domains has thus emerged in order to deal with these non-smooth domains. For instance, it is now well understood to what extend the usual regularity for solutions of classical PDEs extends to Lipschitz domains. It is, in particular, possible to exactly estimate the best regularity for these solutions, which, unlike on smooth domains, will have limited regularity, even if otherwise all the data is smooth. To deal with this "loss of regularity" on Lipschitz domains, new results have emerged in the case of polyhedral domains, a class of domains of great importance in practice. (Many, but not all polyhedral domains are Lipshitz.) In my talk, I will present a general well-posedness result for the Poisson problem with Dirichlet boundary conditions on polyhedral domains in arbitrary dimension. I will also present some applications to optimal convergence rates and some possible extensions to analysis on locally symmetric spaces, on algebraic varieties, and on manifolds arising in General Relativity. The results presented in this talk are based on joint works with B. Ammann, C. Bacuta, A. Ionescu, A. Mazzucato, and L. Zikatanov, and will be accessible to graduate students both in pure and applied mathematics.