The probabilistic approach to the inverse scattering problem is easy to describe: namely, given the results of scattering measurements, to determine the most probable scatterer. We start with a stochastic model of the measurement process in the usual signal-plus-noise form. However, here the signal (i.e., the model of a scattering measurement with no experimental error) is also considered to be random, thereby reflecting our partial lack of a priori knowledge of the nature of the scatterer. In the present treatment, we consider only nonparametric scatterer models (i.e., models involving an essentially infinite number of parameters - at least a number very large compared with the effective number of degrees of freedom in the measurements). We will consider three types of random models of the scatterer: (1) the material property deviations (elastodynamic or electromagnetic) are Gaussian random processes in space, (2) a single property deviation is also a Gaussian random process except for an added positivity bias, and (3) the scatterer has known uniform property deviations in an unknown domain. The inverse scattering problem for all three cases has been solved for the Born approximation using suitable mixtures of analytical and computational approaches. In cases (2) and (3) we employed the conjugate vector technique in order to reduce the computational effort to reasonable size. A special version of case (3), in which internal propagation is negligible, has been treated in the so-called Kirchhoff approximation in the regime of intermediate to high frequencies. A number of results obtained with theoretical synthetic test data will be presented and discussed.