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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.