Author(s): Armin Tavakoli, Alejandro Pozas-Kerstjens, Peter Brown, and Mateus Araújo Sometimes a mathematical tool emerges as uniquely useful and even a defining feature within some branch of physics such as Feynman diagrams. In quantum information theory, the semidefinite program (SDP) has emerged as such a tool. SDP is an optimization task in which a linear objective function is maximized over a set of Hermitian matrices with positive eigenvalues. This review discusses the highly efficient algorithms available for the SDP, and shows how comprehensively the SDP has been deployed in problems of entanglement characterization, quantum nonlocality, quantum channel capacities, and the bounding of ground-state energies. [Rev. Mod. Phys. 96, 045006] Published Wed Dec 04, 2024