This dissertation is concerned with probability density estimation. The main focus is estimation on rotation and motion groups, but some novel methods are developed for Euclidean space as well. First, we propose a new data-driven nonparametric density estimation technique which uses the Voronoi diagram from computational geometry. The technique tessellates the region of interest using the Voronoi diagram and introduces a complete set of basis functions built on the Voronoi diagram. These basis functions are used to describe arbitrary probability densities. By minimizing a properly defined cost function subject to constraints imposed on a weighted sum of the basis functions, the Voronoi estimator can produce a smooth estimate of the underlying density function. We illustrate the performance of this technique by comparing it with other popular methods for univariate density estimation and discuss its extension to the bivariate density. Second, as the major study of this dissertation, we introduce a method which generalizes the Fourier density estimation technique to rotation and motion groups in the context of parametric and nonparametric estimation. This method uses Fourier analysis on Lie groups as a major tool. Lie-group representation theory is used extensively throughout this study. We present the explicit forms of the representation matrices for the groups SO(3), SE(2) and SE(3) in the exponential parameterization. This is accomplished by exponentiating representation matrices of the corresponding Lie algebras. In the case of SE(2) and SE(3) these matrices are infinite dimensional. We study the effects of truncation before exponentiation of the infinite-dimensional matrix representations of se(2) and se(3) to generate approximate representations of SE(2) and SE(3). We demonstrate the nonparametric Fourier density estimation technique using a numerical simulation for the SE(2) case and show how this technique might be used in applications such as robotics. As an example of parametric density estimation on groups, we focus on the Gaussian distribution for SO(3). We provide two methods in detail on how to fit a diffusion model with data sampled from a Gaussian distribution on SO(3). We also provide an efficient method to handle highly concentrated densities centered at the identity by linear approximation of exponential coordinates. This density estimation problem on Lie groups is motivated by applications in many fields, such as robotics, analysis of error in gyroscopic orientation sensors and computing the stiffness matrices of double helical DNA/RNA.