Singularities of continuation of wave fields

The state of the art of the problem of the analytical continuation of wave fields is reviewed. The problem is a multidisciplinary subject which involves radiophysics, acoustics, and optics on the one hand, and mathematical physics and the theory of differential equation, on the other. The qualitative aspects of the problem are examined. A computational algorithm for field singularities is given. The relation between the singularities and the 'computational catastrophes' of scattering models is discussed. To illustrate the theoretical material, numerous examples are discussed in great detail.