We will review the framework of quantum annealing and quantum adiabatic algorithms. We will describe the evidence that they give speed-ups over classical algorithms for optimization problems and the need for non-stoquasticity in the Hamiltonian. We also present some negative results showing evidence that they do not give an exponential speed-up over classical algorithms for these problems. Finally\, we present some recent work on constructing annealing algorithms with non-stoquastic terms. URL:https://wp.physics.wisc.edu/twap/?id=4201 END:VEVENT END:VCALENDAR