In mathematics and applications, quantum annealing is a new method for finding solutions to combinatorial optimization problems and ground states of glassy systems using quantum fluctuations. Quantum fluctuations can be simulated in computers using various quantum Monte Carlo techniques, such as the path integral Monte Carlo method, and thus they can be used to obtain a new kind of heuristic algorithm for global optimization. It can be said that the idea of quantum annealing comes from the celebrated classical simulated thermal annealing invented by Kirkpatrick. However, unlike a simulated annealing algorithm, which utilizes thermal fluctuations to help the algorithm jump from local optimum to global optimum, quantum annealing algorithms utilize quantum fluctuations to help the algorithm tunnel through the barriers directly from local optimum to global optimum. According to the previous studies, although the quantum annealing algorithm is not capable, in general, of finding solutions to NP-complete problems in polynomial time, quantum annealing is still a promising optimization technique, which exhibits good performances on some typical optimization problems, such as the transverse Ising model and the traveling salesman problem. Provided in this paper is an overview of the principles and research progresses of quantum annealing algorithms in recent years; several different kinds of quantum annealing algorithms are presented in detail; both the advantages and disadvantages of each algorithm are analyzed; and prospects for the research orientation of the quantum annealing algorithm in future are given.