In these lectures we shall describe some recent results concerning the spectral theory of general non-self-adjoint elliptic boundary value problems. We shall be interested in the following problems: (i) Completeness of eigenfunctions, (ii) Angular distribution of eigenvalues, (iii) Asymptotic distribution of eigenvalues. The general plan of the lectures is as follows. In Lecture I we shall introduce the general class of regular elliptic boundary value problems and discuss the growth of certain resolvents in the complex plane. In Lecture II we shall establish completeness results for eigen-functions of general elliptic problems obtaining also some results on the angular distribution of eigenvalues. In Lecture III we shall discuss some special classes of elliptic problems such as self-adjoint problems and absolutely elliptic problems. In Lecture IV we shall describe a very general result on the asymptotic distribution of eigenvalues of non-self-disjoint elliptic problems.