To achieve an improvement of plasma confinement by an effective edge plasma control, the local island divertor (LID) was originally proposed in the National Institute for Fusion Science in the early 1980s. The LID is a kind of island divertor that utilizes the island separatrix as the channeling magnetic structure of the divertor, and it has the particular characteristic of localizing the particle recycling in very small areas. Thus, it is possible to construct a compact closed divertor configuration with efficient pumping capability, which results in the low-recycling condition in the edge region. In this paper the LID project is reviewed, from the physics design phase with numerical validation or estimation of the LID principle to a recent experimental result of the superdense core mode, which is a promising discharge for next-generation devices.