Gradient-index (GRIN) materials provide interesting ways to direct light propagation inside a bulk medium. Their application in optical systems as compact optical elements offer many advantages such as convenient form factor, unique dispersion characteristics, aberration correction capabilities, etc. With the recent technological advances in the fabrication techniques for these materials, it is reasonable to speculate that arbitrary refract index distributions in GRIN media can be realized in the near future. The integration of GRIN components into optical systems requires accurate knowledge of their refractive index distribution. Numerical methods for recovering the refractive index of the material using boundary value measurements of position and slope for interrogating rays that transit the medium are described. For one-dimensional index profiles, we develop a bootstrap algorithm for recovering the refractive index in successive regions of the overall profile from the boundary value data. We then compare the reconstructed index profile obtained in this method with that of a different method based on ray displacement and show good agreement in computer simulation as well as in experimental measurement. In the case of two-dimensional refractive index distributions, we show that the path integrals describing beam deflection inside the material can be cast in the form of linear algebraic equations using a simplifying assumption that decouples unknown ray trajectories inside the medium from the refractive index. The resulting linear system is inverted numerically to recover the refractive index distribution, and the ray trajectories are subsequently ascertained through an iterative ray trace procedure. Using boundary values of ray position and slope generated from a numerical ray trace, we show that this method can achieve RMS index errors less than 0.5% of the refractive index range. In addition, we explore the application of GRIN components in designing optical resonators. Using a Green's function approach, we show that wave propagation inside GRIN media follows the Huygens-Fresnel principle and can be calculated from the superposition of secondary wavelets. A design procedure for achieving coherent mode conversion in GRIN media is described, and a tool for analyzing optical resonators employing an intracavity GRIN component is developed. We use this tool to calculate the spatial eigenmodes of a flat-mirror resonator employing a Gaussian-to-flat-top GRIN mode converter and determine its modal properties.