The lectures will give an introduction to the theory and applications of convex optimization, and an overview of recent developments in algorithms. The first lecture will cover the basics of convex analysis, focusing on the results that are most useful for convex modeling, i.e., recognizing and formulating convex optimization problems in applications. We will introduce conic optimization, and the two most widely studied types of conic optimization problems, second-order cone and semidefinite programs. The material will be illustrated with applications to robust optimization, convex relaxations in nonconvex optimization, and convex techniques for sparse optimization. Lecture 2 will cover interior-point methods for conic optimization, including path-following methods and symmetric primal-dual methods, and the numerical implementation of interior-point methods. Lecture 3 will focus on first-order algorithms for large-scale convex optimization, including recent developments in the area of proximal gradient methods, and on dual decomposition and multiplier methods.