The purpose of the inverse-transport problems that we address is to reconstruct the material distribution inside an unknown object undergoing a nondestructive evaluation. We assume that the object is subjected to incident beams of photons or particles and that the exiting radiation is measured with detectors around the periphery of the object. In the present work we focus on problems in which radiation can undergo significant scattering within the optically thick object. We develop a set of reconstruction strategies to infer the material distribution inside such objects. When we apply these strategies to a set of neutron-tomography test problems we find that the results are substantially superior to those obtained by previous methods. We first demonstrate that traditional analytic methods such as filtered back projection (FBP) methods do not work for very thick, highly scattering problems. Then we explore deterministic optimization processes, using the nonlinear conjugate gradient iterative updating scheme to minimize an objective functional that characterizes the misfits between forward predicted measurements and actual detector readings. We find that while these methods provide more information than the analytic methods such as FBP, they do not provide sufficiently accurate solutions of problems in which the radiation undergoes significant scattering. We proceed to present some advances in inverse transport methods. Our strategies offer several advantages over previous reconstruction methods. First, our optimization procedure involves the systematic use of both deterministic and stochastic methods, using the strengths of each to mitigate the weaknesses of the other. Another key feature is that we treat the material (a discrete quantity) as the unknown, as opposed to individual cross sections (continuous variables). This changes the mathematical nature of the problem and greatly reduces the dimension of the search space. In our hierarchical approach we begin by learning some characteristics of the object from relatively inexpensive calculations, and then use knowledge from such calculations to guide more sophisticated calculations. A key feature of our strategy is dimension-reduction schemes that we have designed to take advantage of known and postulated constraints. We illustrate our approach using some neutron-tomography model problems that are several mean-free paths thick and contain highly scattering materials. In these problems we impose reasonable constraints, similar to those that in practice would come from prior information or engineering judgment. Our results, which identify exactly the correct materials and provide very accurate estimates of their locations and masses, are substantially better than those of deterministic minimization methods and dramatically more efficient than those of typical stochastic methods.