[en] The WKB approximation to the one-particle Schrodinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth ''background'' term (single reflection approximation) and a second resonating term. The complex solutions of the classical equations of motion describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases property of classical paths is discussed. The general pattern of the trajectories depends only on the product epsilon=E Theta, and not on energy and angle separately. This property is confirmed by experiments. Finally, a general classification of the different types of elastic heavy ion cross sections is given