In this book, we focus our attention on elliptic boundary value problems in domains with nonsmooth boundaries and problems with mixed boundary conditions. So far this topic has been mainly ignored. Indeed most of the available mathematical theories about elliptic boundary value problems deal with domains with very smooth boundaries; few of them deal with mixed boundary conditions. However, the majority of the elliptic boundary value problems which arise in practice are naturally posed in domains whose geometry is simple but not smooth. These domains are very often three-dimensional polyhedra. For the purpose of solving them numerically these problems are usually reduced to two-dimensional domains. Thus the domains are plane polygons and the boundary conditions are mixed. Accordingly this book is primarily intended for mathematicians working in the field of elliptic partial differential equations as well as for numerical analysts and users of such elliptic equations.