Explore quantum algorithms for optimization problems in this comprehensive lecture that introduces key concepts and algorithms developed for solving optimization challenges on quantum computers. Learn about quantum optimization algorithms inspired by physics including adiabatic algorithms, variational algorithms, QAOA, and quantum annealing, while examining their fundamental limitations. Discover quantum algorithms for convex optimization that accelerate first- and second-order methods and address oracular problems. Investigate applications to combinatorial optimization including graph problems and quadratic binary optimization. Gain insights into the potential and constraints of quantum computing approaches to optimization from the early developments in the field to current methodologies. This lecture was recorded during the CEMRACS Summer School on Quantum Computing at the Centre International de Rencontres Mathématiques in Marseille, France.