In this thesis, we propose a method to robustly estimate the parameters that controls the mapping of a shape (model shape) onto another (target shape). The shapes of interest are contours in the 2D space, surfaces in the 3D space and point clouds (either in 2D and 3D spaces). We propose to model the shapes using Gaussian Mixture Models (GMMs) and estimate the transformation parameters by minimising a cost function based on the Euclidean (L2) distance between the target and model GMMs. This strategy allows us to avoid the need for the computation of one to one point correspondences that are required by state of the art approaches making them sensitive to both outliers and the choice of the starting guess in the algorithm used for optimisation. Shapes are well represented by GMMs when careful consideration is given to the design of the covariance matrices. Compared to isotropic covariance matrices, we show how shape matching with L2 can be made more robust and accurate by using well chosen non isotropic ones. Our framework offers a novel extension to L2 based cost functions by allowing prior information about the parameters to be included. Our approach is therefore fully Bayesian. This Bayesian-L2 framework is tested successfully for estimating the affine transformation between data sets, for fitting morphable models and fitting ellipses. Finally we show how to extend this framework to shapes defined in higher dimensional feature spaces in addition to the spatial domain. TARA (Trinity's Access to Research Archive) has a robust takedown policy. Please contact us if you have any concerns: rssadmin@tcd.ie