Thesis (Ph.D., Physics)--University of Idaho, June 2014 | Nucleon-nucleon scattering is a most fundamental process in nuclear physics. From the theoretical standpoint, its description in momentum space involves the solution of an integral equation in three dimensions, which is typically accomplished with the help of a partial wave expansion of the scattering amplitude. It is the purpose of this work to present a method for solving the nucleon-nucleon scattering equation without the use of such expansion. After verifying the accuracy of our numerical tools by comparing with existing solutions of the nucleon-nucleon scattering amplitude in free space, we proceed to apply the method to the equation describing scattering of two nucleons in the nuclear medium, known as the Bethe-Goldstone equation. An important feature of this equation is the presence of the so-called "Pauli blocking operator", which prevents scattering of two fermions into occupied states, as required by the Pauli principle. In standard solution methods based on partial wave expansions, it is necessary to apply an approximation to this operator, which involves averaging over angular variables and is therefore known as the "spherical approximation". In our method, this approximation can be avoided. Thus, a focal point of this study is a comparison of Pauli blocking effects calculated in the (angle-dependent) three-dimensional formalism as compared to the usual spherical approximation. We present results for nucleon-nucleon amplitudes and observables and discuss their implications.