In this thesis we compare various methods for estimating the unknown parameters in mixtures of circular and spherical distributions. We study the von Mises distribution on the circle and the Fisher distribution on the sphere. We propose a new method of estimation based on the characteristic function and compare it with the classical methods based on maximum likelihood and moments. Thus far these methods have only been successfully applied to distributions on the line. Here we show that the extension to circular and spherical distributions is reasonably straightforward and convergence to the final estimates is fairly rapid. We apply these methods to various simulated and real data sets and show that the results obtained for the mixture of two von Mises distributions are satisfactory but generally depend on the sample size and method of estimation used. However, results obtained for the mixture of two Fisher distributions show that maximum likelihood performs best overall.