The course aims at providing Ph.D. students with analytical tools useful in applied research on general wave phenomena. The unifying theme is that of complex analysis, of which a compact, self-contained introduction is presented. Fundamental techniques for the asymptotic evaluation of integrals are then illustrated, including the Laplace and saddle-point methods. Applications are focused on the analysis of time-harmonic waves excited in planar layered structures by canonical sources and on scattering of plane waves from half planes. As concerns the former, different wave species will be defined and physically discussed (space waves, surface waves, leaky waves, lateral waves). As concerns the latter, the Wiener-Hopf method will be introduced.