The main goal of this thesis is to solve numerically the exterior and interior Robin boundary value problems via a boundary integral equation method, which has an advantage of decreasing the computational dimension of the problem. Representing the solution by a layer potential, we reduce the dierential problem in a bounded and an unbounded domain to the Fredholm integral equation of the second kind over the boundary. In the case of exterior problem in two dimension, the fundamental solution to the Laplace equation is logarithmic, and hence additional condition or modification has to be applied that keeps the solution bounded in the unbounded domain. Instead of using a classical singlelayer potential and enforcing a condition on the unknown density we propose a modified single layer potential approach. After investigating uniqueness and existence of solution to the obtained integral equations of second kind, we solve the equations numerically by the Nyström method. For the numerical integration of integral operators with continuous kernels the trigonometric quadratures on an equidistant mesh is used. For the numerical integration of weakly singular kernels we first splitt o the logarithmic singularity and apply a special quadrature rule for the improper integrals. The feasibility of the proposed methods, covergence order (super-algebraic for smooth data) is illustrated by numerical examples.