I have been asked to give two talks [Walker-Ames Lecture at the University of Washington, June 18, 1979] on recent developments in branches of my main interest. Unfortunately, I have not been able to follow the most recent literature, but I believe it is worthwhile to recognize the essential features of modern computation techniques by confronting them with older ones; this might be achieved by a historical survey, to be given in particular in my first talk, while the second one will emphasize the common aspects of procedures which in the literature have been worked out only for one special branch of physics or technology but are just as well applicable to other related fields. This division requires, moreover, that my first talk almost exclusively concern phenomena the essential properties of which already follow from the effects produced by a single infinitesimal source, while the second lecture considers the consequences of the detailed structure of the source in view of its finite dimensions. Thus let me begin with a discussion of the propagation of terrestrial radio waves, a problem the development of which I have followed with great interest during my scientific career. Represented in its simplest idealized form, we here want to know the electromagnetic field of a point source transmitter situated near the earth's surface and situated there in the presence of a homogeneous, spherical earth surrounded by an also homogeneous medium, the atmosphere. The interest in this diffraction problem arose when, in 1902, Marconi succeeded for the first time in transmitting damped radio waves across the Atlantic Ocean. Although this success was in the beginning mainly viewed from the economical side (possible future competition with telecommunication using sea cables), a bit later scientists wondered about its physical aspect, since here the curvature of the earth excluded rectilinear propagation; indeed the latter should take place along a chord through the earth, which, however, would involve an extremely high attenuation. The question then became whether the atmosphere really behaves as an almost homogeneous medium or whether just its small inhomogeneity might make possible the transmission over large distances along curved rays, in the sense of geometrical optics. We should not forget that the ionosphere had yet to be discovered at the time, although as early as 1878 the American BaIfour Stewart had introduced a model of electric currents at high altitudes in the atmosphere in order to explain the part of the earth's magnetic field that had to be ascribed (according to a theory of Gauss) to a source outside the earth. Kennelly, not conscious of Stewart's model, then put forward the hypothesis of some reflecting layer, while Heaviside believed that oceans might guide waves, in view of their electric conductivity. According to Heaviside, however, the high conductivity of a rarefied gas (large mean free paths) at high altitudes might also play a role, though only a secondary one. In honor of these two scientists the name Kennelly-Heaviside layer became customary a long time before the present term ionosphere. Since all these assumptions could not be verified at the time, pure mathematicians preferred to consider first the simplest model, that of pure diffraction around a completely homogeneous body of the earth surrounded by an also completely homogeneous atmosphere; this then constitutes the idealized mode I mentioned at the beginning.