Several direct and inverse problems involving inhomogeneous media are studied in this dissertation. First we study the problem of constructing the pressure field excited by a timeharmonic point source in a shallow ocean with an interactive seabed. Both constant coefficient cases and variable coefficient case, elastic seabed and poro-elastic seabed are considered. A general procedure for solving this kind of problems is developed. The development of this procedure is based on the methods of integral transformation, Mitta-Leffler decomposition, generalized Fourier expansions, transmutation, symbolic and numerical computation. The second problem is to compute numerically the pressure field scattered by an object in a shallow ocean with an elastic seabed. All coefficients are assumed to be constant. We use the boundary integral equation method based on the fundamental singular solution for such a wave guide. We also discuss the algorithm for numerical computations, especially, for parallel computing. The algorithm is implemented on a parallel computer. The first inverse problem we study is how to reconstruct the radially dependent coefficient in a potential equation and a steady heat equation from a single pair of DirichletNeumann boundary data. A solution to this problem is derived by studying the forward problem thoroughly. A transmutation for inhomogeneous boundary data and with a singular kernel is used for constructing the solution to the forward problem in three different cases. Then the inverse problem is reduced to easier sub-problems. The last part of this dissertation is devoted to investigation of the inverse problem of reconstructing two radially dependent Lam'e coefficients of a ball simultaneously. This is a one-dimensional, multiple unknown, undetermined coefficient problem. A metric form transmutation is developed to represent the solution to the forward problem, which then enables us to determine the coefficients uniquely by suing only two sets of boundary data.