A large number of machine learning algorithms are critically dependent on the underlying distance/metric/similarity function. Learning an appropriate distance function is therefore crucial to the success of many methods. The class of distance functions that can be learned accurately is characterized by the amount and type of supervision available to the particular application. In this thesis, we explore a variety of such distance learning problems using different amounts/types of supervision and provide efficient and scalable algorithms to learn appropriate distance functions for each of these problems. First, we propose a generic regularized framework for Mahalanobis metric learning and prove that for a wide variety of regularization functions, metric learning can be used for efficiently learning a kernel function incorporating the available side-information. Furthermore, we provide a method for fast nearest neighbor search using the learned distance/kernel function. We show that a variety of existing metric learning methods are special cases of our general framework. Hence, our framework also provides a kernelization scheme and fast similarity search scheme for such methods. Second, we consider a variation of our standard metric learning framework where the side-information is incremental, streaming and cannot be stored. For this problem, we provide an efficient online metric learning algorithm that compares favorably to existing methods both theoretically and empirically. Next, we consider a contrasting scenario where the amount of supervision being provided is extremely small compared to the number of training points. For this problem, we consider two different modeling assumptions: 1) data lies on a low-dimensional linear subspace, 2) data lies on a low-dimensional non-linear manifold. The first assumption, in particular, leads to the problem of matrix rank minimization over polyhedral sets, which is a problem of immense interest in numerous fields including optimization, machine learning, computer vision, and control theory. We propose a novel online learning based optimization method for the rank minimization problem and provide provable approximation guarantees for it. The second assumption leads to our geometry-aware metric/kernel learning formulation, where we jointly model the metric/kernel over the data along with the underlying manifold. We provide an efficient alternating minimization algorithm for this problem and demonstrate its wide applicability and effectiveness by applying it to various machine learning tasks such as semi-supervised classification, colored dimensionality reduction, manifold alignment etc. Finally, we consider the task of learning distance functions under no supervision, which we cast as a problem of learning disparate clusterings of the data. To this end, we propose a discriminative approach and a generative model based approach and we provide efficient algorithms with convergence guarantees for both the approaches. / text