This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to simplify and to algebraize the index theory by means of pseudo-differential operators and new methods in the spectral theory of matrix polynomials. This latter theory provides important tools that will enable the reader to work efficiently with the principal symbols of the elliptic and boundary operators. It also leads to important simplifications and unifications in the proofs of basic theorems such as the reformulation of the Lopatinskii condition in various equivalent forms, homotopy lifting theorems, the reduction of a system with boundary conditions to a system on the boundary, and the index formula for systems in the plane. . This book is suitable for use in graduate level courses on partial differential equations, elliptic systems, pseudo-differential operators, and matrix analysis. All the theorems are proved in detail, and the methods are well illustrated through numerous examples and exercises.