This talk is concerned with the numerical computation of  transmission eigenvalues in the inverse scattering theory, which are shed light on the material properties of scattering object. The analysis of that problem cannot be covered by standard theory of elliptic partial differential equations since it is neither elliptic nor self-adjoint.  A novel integral equation formulation built upon the Schur complement to a two by two system of boundary integral equations is used to reformulate the problems of transmission eigenvalues. The Nystrom discretization is then employed to obtain an eigenvalue problem for a matrix. At the last step, the corresponding matrix eigenvalue problem is computed by the recursive integral method.