<p dir="ltr">In this thesis, we consider the inverse problem of reconstructing the shape, position, and size of an unknown scattering object. We will talk about different methods used for nondestructive testing in scattering theory. We will consider qualitative reconstruction methods to understand and determine important information about the support of unknown scattering objects. We will also discuss the material properties of the system and connect them to certain crucial aspects of the region of interest, as well as develop useful techniques to determine physical information using inverse scattering theory. </p><p><br></p><p dir="ltr">In the first part of the analysis, we consider the transmission eigenvalue (TE) problem associated with the scattering of a plane wave for an isotropic scatterer. In particular, we examine the transmission eigenvalue problem with two conductivity boundary parameters. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter, whereas we will consider the case of two parameters. We will prove the existence and discreteness of the transmission eigenvalues. In addition, we will study the dependence of the TE's on the physical parameters and connect the first transmission eigenvalue to the physical parameters of the problem by a monotone-type argument. Lastly, we will consider the limiting procedure as the second boundary parameter vanishes at the boundary of the scattering region and provide numerical examples to validate the theory presented in Chapter 2. </p><p><br></p><p dir="ltr">The connection between transmission eigenvalues and the system's physical parameters provides a way to do testing in a nondestructive way. However, to understand the region of interest in terms of its shape, size, and position, one needs to use different techniques. As a result, we consider reconstructing extended scatterers using an analogous method to the Direct Sampling Method (DSM), a new sampling method based on the Landweber iteration. We will need a factorization of the far-field operator to analyze the corresponding imaging function for the new Landweber direct sampling method. Then, we use the factorization and the Funk--Hecke integral identity to prove that the new imaging function will accurately recover the scatterer. The method studied here falls under the category of qualitative reconstruction methods, where an imaging function is used to retrieve the scatterer. We prove the stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.</p><p><br></p><p dir="ltr">Motivated by the work done with the transmission eigenvalue problem with two conductivity parameters, we also study the direct and inverse problem for isotropic scatterers with two conductive boundary conditions. In such a problem, one analyzes the behavior of the scattered field as one of the conductivity parameters vanishes at the boundary. Consequently, we prove the convergence of the scattered field dealing with two conductivity parameters to the scattered field dealing with only one conductivity parameter. We consider the uniqueness of recovering the coefficients from the known far-field data at a fixed incident direction for multiple frequencies. Then, we consider the inverse shape problem for recovering the scatterer for the measured far-field data at a fixed frequency. To this end, we study the direct sampling method for recovering the scatterer by studying the factorization for the far-field operator. The direct sampling method is stable concerning noisy data and valid in two dimensions for partial aperture data. The theoretical results are verified with numerical examples to analyze the performance using the direct sampling method. </p>