The theory of pseudo-differential operators has been developed as a powerful tool to handle particial differential equqtions. Here the pseudo-differential operators, and more generally the Fourie integral operators, include as special cases both the differential operators, their solution operators (integral operators), and compositions of these types. For equations on manifolds with boundary, Eskin, Vishik and Boutet de Monvel invented in particular the calculus of pseudo-differential boundary operators, that applies to elliptic boundary value problems. The aim of this book is to develop a functional calculus for such operators; i.e. to find the structure and properties of functions of these operators defined abstractly by functional analysis.