We tried to summarize the status of the laser fusion research in the world as the beginning of this review series. The mainstream of laser fusion research is the central ignition scheme. We explain processes to realize the fusion ignition by the central ignition scheme and then describe briefly progresses of the central ignition studies in the United States, Europe/UK, and China. In addition, we introduce progresses of the fast-ignition method, which aims to realize fusion energy by avoiding the barriers of the central ignition scheme.