Compressive Sensing has recently had a tremendous impact in science and engineering, because it revealed the theoretical possibility of acquiring structured highdimensional objects using much less information than previously expected, and more importantly because it provided practical procedures to do so. The foundations of the field rely on an elegant mathematical theory that has matured over the past few years. The goal of the lecture series is to highlight the main aspects of this theory. In particular, the following items will be discussed: the exact reconstruction of sparse vectors; the stable and robust reconstruction of nearly sparse vectors acquired with minor errors; the basic reconstruction algorithms (linear and convex programming, greedy algorithms, thresholdingbased strategies); the coherence of a matrix; its restricted isometry constants; random measurement matrices; their optimality; and structure beyond sparsity.