The Birman-Schwinger principle is one of the standard tools in spectral analysis of self-adjoint Schrödinger operators. This useful technique allows to reduce the eigenvalue problem for the Schrödinger operator $-\Delta +V$ to an eigenvalue problem involving a sandwiched resolvent of the unperturbed operator $-\Delta$. In this talk we first review the classical Birman-Schwinger principle and illustrate it with some typical applications in spectral analysis. Afterwards we discuss some recent extensions for the characterization of the generalized eigenvectors of non-selfadjoint Schrödinger operators and other general non-selfadjoint second order elliptic differential operators. The talk is based on a joint work with A.F.M. ter Elst and F. Gesztesy.