Explore quantum optimization algorithms and their applications in this comprehensive lecture that delves into the intersection of quantum computing and optimization problems. Learn about quantum algorithms inspired by physics, including adiabatic algorithms, variational algorithms, QAOA, and quantum annealing, while understanding their fundamental limitations. Discover quantum approaches to convex optimization through acceleration of first- and second-order methods and oracular problems. Examine practical applications to combinatorial optimization, focusing on graph problems and quadratic binary optimization. Gain insights into the theoretical foundations and practical implementations of quantum optimization techniques that have been developed since the early days of quantum computing. This lecture forms part of the CEMRACS Summer School on Quantum Computing, providing advanced mathematical perspectives on how quantum computers can potentially solve complex optimization challenges across various domains.