Several topics in the realm of Nuclear Magnetic Resonance (NMR) spectroscopy are discussed in this dissertation. These topics can be divided into two major sections. The first section considers the treatment of data from standard NMR relaxation and diffusion experiments. The current standard for the analysis of transients in low-field NMR is the Inverse Laplace Transform (ILT). Often leading to an inconsistency in the stability of results, the ILT tends to be an ill-posed problem like many other inversion methods. Alternative data processing methods have recently been introduced in an attempt to improve the resolution, stability, and accuracy of results for both discrete and continuous sets of recovered constants. The application of the Matrix Pencil Method (MPM), a generalized eigenvalue-based algorithm, to NMR transients has recently gained traction as a reliable processing method with low computational costs. Here, the MPM is first extended to the resolution and quantitative analysis of multiple longitudinal (T1) relaxation components from data acquired by a stray-field sensor. A comparison of MPM to ILT is conducted by testing several combinations of Gd3+-doped 0.9% saline samples with an array of concentrations. It is shown that MPM not only has a greater ability to resolve than ILT at low SNR, but the resulting time constants and relative component weights are also closer to their expected values. The results of the stray-field study brings about two questions: What are the true limitations of the MPM resolution capabilities and how can experiments be optimized to gain the most information possible from the data? The MPM provides exact solutions for noise-free transients with few discrete components. These solutions increasingly deviate as the SNR of the transient decreases. Numerical and analytical methods are developed to use this knowledge at a given SNR to predict the prime sampling interval to obtain the maximum resolution and accuracy possible. The second major section concerns the effects of spatial confinement on diffusion and chemical exchange. Two-site exchange has been exhaustively studied and its dynamics are well understood. Three-site exchange, on the other hand, is a drastically different story. Recent kinetics studies on three-site systems have revealed asymmetric exchange behavior when subjected to restrictive environments. This asymmetry is indicative of a violation of detailed balance which states that all pairs of relaxation sites should experience equivalent exchange at thermal equilibrium. An investigation on the effects of confinement on three-site exchange is conducted using two 2D numerical methods, a Monte-Carlo vacancy diffusion simulation and a molecular dynamics gas diffusion simulation with elastic particles. These simulations reveal that this asymmetry indeed arises when the diffusive motion is driven away from standard Brownian dynamics. Under this driven equilibrium, a circular flux between relaxation sites develops which goes against detailed balance. The cyclical diffusive behavior potentially arises from the excitation of the diffusive eigenmodes of a pore.