The physical motivations and performance of the TAIGA (Tunka Advanced Instrument for cosmic ray physics and Gamma Astronomy) project are presented. The TAIGA astrophysical complex addresses ground-based gamma-ray astronomy at energies from a few TeV to several PeV and cosmic ray physics from 100 TeV to several EeV and astroparticle physics. The pilot TAIGA-1 complex is located in the Tunka valley, ~50 km west of the southern tip of lake Baikal. It includes a timing Cherenkov TAIGA-HiSCORE array with 120 wide-angle optical stations distributed over an area of about 1 square kilometer and three 4-m class Imaging Atmospheric Cherenkov Telescopes of the TAIGA-IACT array. The latter array has a shape of a triangle with side lengths of about 300m, 400m and 500m. There are three approaches to selecting gamma quanta from the hadron background in the TAIGA experiment: (1) IACT operation in standalone mode; (2) Stereo mode operation of two or more IACTs; (3) Hybrid mode operation -joint operation of TAIGA-HiSCORE and IACTs. The main advantage of the hybrid operation of the IACTs and timing is their good gamma/hadron separation, even with only a few telescopes on the large area. Present status of the project, the first experimental results and plans for the future 10-20 km^2 TAIGA are presented.