A critical review is presented of recent progress in classical diffraction theory. Both scalar and electromagnetic problems are discussed. The report may serve as an introduction to general diffraction theory although the main emphasis is on diffraction by plane obstacles. Various modifications of the Kirchhoff and Kottler theories are presented. Diffraction by obstacles small compared with the wavelength is discussed in some detail. Other topics included are: variational formulation of diffraction problems, the Wiener-Hopf technique of solving integral equations of diffraction theory, the rigorous formulation of Babinet's principle, the nature of field singularities at sharp edges, the application of Mathieu functions and spheroidal wave functions to diffraction theory. Reference is made to more than 500 papers published since 1940.