This work deals with binary classiﬁcation in a sparse high-dimensional model. Essen tially, we have two predetermined subclasses of the normally distributed population and we want to assign a new observation to one of the two subclasses. In this thesis, we look at a classiﬁcation problem in which the sample size is less than the dimension of the data. The goal is to ﬁnd good methods for classiﬁcation that can minimize our chances of misclassiﬁcation under these circumstances. A typical high-dimensional problem, like the one studied in this thesis, has a large number of unknown parameters and not enough data to make reliable inferences about these parameters. The classical statistical methods were not designed to cope with high-dimensional problems. Therefore, when studying the classiﬁcation problem of our interest, we have proposed original statistical ideas and tools based on the notion of ‘sparsity’.