The object of this book is to study elliptic boundary value problems with data, which may have singularities (e.g. mixed boundary value problems, boundary value problems with transmission property and singular crack problems, a prominent example being the Zaremba problem for the Laplace equation with mixed Dirichlet and Neumann boundary conditions). The authors’ aim is to study such problems based on the general calculus of operators on a manifold with edges or conical singularities and boundary. General theory then allows the construction of parametrix for considered problems and the proof of the regularity of the solutions. In the last chapter, an applied approach with operator algebras and symbolic structures on manifolds with singularities is discussed, together with motivations and branches of research over the past years and new challenges and open problems for the future. The mathematicians and physicists interested in elliptic differential equations with singularities will definitely appreciate the unified approach presented in the book. The authors also address the text to advanced students and specialists working in the field of analysis on manifolds with geometric singularities, applications of index theory and spectral theory, operator algebras with symbolic structures, quantization and asymptotic analysis.