Convex optimization has emerged as useful tool for applications that include data analysis and model fitting, resource allocation, engineering design, network design and optimization, finance, control and signal processing, and circuit sizing. After an overview of the mathematics, algorithms, and software frameworks for convex optimization, we turn to common themes that arise across applications, such as sparsity and relaxation. We describe recent work on real-time embedded convex optimization, in which small problems are solved repeatedly and reliably in millisecond or microsecond time frames, with growing applications in control and resource allocation.