Introduction. Boundary Elements and Finite Elements. Historical development of the BEM. Preliminary Mathematical Concepts. The Gauss-Green theorem. The divergence theorem of Gauss. Green’s second identity. The Dirac delta function. The BEM for Potential Problems in Two Dimensions. Fundamental solution. The direct BEM for the Laplace and the Poisson equation. Transformation of the domain integrals to boundary integrals. The Dual reciprocity method. The BEM for potential problems in anisotropic bodies. Numerical Implementation of the ΒΕΜ. The BEM with constant boundary elements. Programming of the method in FORTRAN. Multiply connected domains. The method of subregions. Boundary Element Technology. Linear elements. Higher order elements. Applications. Torsion of non-circular bars. Deflection of elastic membranes. Heat transfer problems. Fluid flow problems. The BEM for the plate problem. The Rayleigh-Green identity for the biharmonic operator. Fundamental solution. Integral representation of the solution. The boundary integral equations. Two-Dimensional Elastostatic Problems. Equations of plane elasticity. Betti’s reciprocal identity. Fundamental solution of the Navier equations. Integral representation of the solution.