The solution of elliptic partial differential equations of second order can be described by the complete Cauchy data when a fundamental solution is known. To find the complete Cauchy data appropriate boundary integral equations have to be formulated and analyzed. In particular, the unique solvability has to be insured. We consider the Laplace and the Helmholtz equation, the system of linear elastostatics and the Stokes system, and the Maxwell equations. Applications are numerous,
e.g. in electromagntics and in mechanics.