  In many Direct and Inverse Scattering problems one has to use a
parameter-fitting procedure, because analytical inversion procedures are often
not available. In this paper a variety of such methods is presented with a
discussion of theoretical and computational issues. The problem of finding
small subsurface inclusions from surface scattering data is solved by the
Hybrid Stochastic-Deterministic minimization algorithm. A similar approach is
used to determine layers in a particle from the scattering data. The Inverse
potential scattering problem for spherically symmetric potentials and
fixed-energy phase shifts as the scattering data is described. It is solved by
the Stability Index Method. This general approach estimates the size of the
minimizing sets, and gives a practically useful stopping criterion for global
minimization algorithms. The 3D inverse scattering problem with fixed-energy
data and its solution by the Ramm's method are discussed. An Obstacle Direct
Scattering problem is treated by a Modified Rayleigh Conjecture (MRC) method. A
special minimization procedure allows one to inexpensively compute scattered
fields for 2D and 3D obstacles having smooth as well as nonsmooth surfaces. A
new Support Function Method (SFM) is used for Inverse Obstacle Scattering
problems. Another method for Inverse scattering problems, the Linear Sampling
Method (LSM), is analyzed.
