Nanoindentation is a widely recognized method for characterizing the mechanical properties of thin films and small volumes. This dissertation reports the results of finite element analyses of elastic and elastic-plastic indentation by a rigid cone aimed at improving methods for measuring of contact area, hardness and elastic modulus by nanoindentation methods. Analytical and finite element results are presented which show that corrections to Sneddon's solution are needed to properly describe elastic indentation by a cone. Since most nanoindentation methods are based on Sneddon's solution, these corrections have important consequences for making accurate mechanical property measurements. Elastic-plastic finite element simulations are presented which show that pile-up can significantly affect the accuracy of nanoindentation measurements. It is shown that an experimentally measurable parameter, the ratio of the final depth to the total depth of indentation, is useful in determining the amount of pile-up in the material. An investigation of plastic zones and stresses in indented materials reveals important correlations between them and the nanoindentation behavior of the material. Implications of these results for indentation cracking are also discussed. A long standing problem in nanoindentation is why load-displacement data obtained during unloading fit well to a power relation with a power law exponents in the range 1.25-1.50. Finite element simulations combined with elastic contact analyses are presented which provide a simple explanation for this behavior. General recommendations are made for improving of nanoindentation methods for measuring mechanical properties.