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In this survey paper we report on recent developments of the hp-version of the boundary element method(BEM).As model problems we consider weakly singular and hypersingular integral equations of the first kind on a planar,open surface.We show that the Galerkin solutions (computed with the hp-version on geometric meshes) converge exponentially fast towards the exact solutions of the integral equations.An hp-adaptive algorithm is given and the implementation of the hp-version BEM is discussed together with the choice of efficient preconditioners for the ill-conditioned boundary element stiffness matrices.We also comment on the use of the hp-version BEM for solving Signorini contact problems in linear elasticity where the contact conditions are enforced only on the discrete set of Gauss-Lobatto points.Numerical results are presented which underline the theoretical results.