The status of the TAIGA experiment (Tunka Advanced Instrument for cosmic-ray physics and Gamma-ray Astronomy) located in the Tunka Valley is presented. The paper presents mainly the tasks, developed approaches for their solution, and first results on high-energy gamma-ray astronomy (10 TeV and higher) obtained from a two- to three-year exposure. The current tasks of gamma-ray astronomy and plans for development of the installation are discussed. [ABSTRACT FROM AUTHOR]