While this is by no means an exhaustive review, we have overviewed a number of efective methods for representing data sparsely, which is to say, solving \(min_a \left\|x-Ba\right\|\) with sparsity constraints on \(a\) and related problems. We started in chapter 1 by going over methods most widely used in modern signal processing, such as the DCT and DFT, and then introduced some general-purpose methods for sparse coding - namely matching pursuit and basis pursuit. We touched on some of the problems inherent in sparse coding (the fact that the base problem is NP-hard), and how to circumvent those issues. In chapter 2, we introduced a number of methods for not only optimizing the sparse representation, but the dictionary \(B\) as well, and sometimes simultaneously. By discussing k-means clustering and singular-value decompositions, we built the knowledge base necessary to understand the K-SVD algorithm, and we tested it on some synthetic data. We also briefly overviewed another approach with broad applications - the principal component analysis, necessary to understand MPCA discussed later on. In the last chapter, chapter 3, we enter the domain of tensors. This is a natural step, as much real-world data, especially visual data, is more naturally represented as tensors of higher orders instead of vectors or matrices. We went over some basic tensor arithmetic, and then reviewed two methods. One is a dictionary learning method built to analyze dynamic textures, and the other is multilinear principal component analysis - a generalization of PCA to tensor space. The applications of sparse coding and dictionary learning are vast, and new methods are being developed rapidly in order to deal with the modern problem of ever more data being accumulated, sent over communication channels and needing to be analyzed or compressed