The DESI experiment will put tight constraints on the dark energy with the observation of more than 40 million spectroscopic redshifts, mostly at z < 2. It started its Main Survey in May, 2021, making it the first Stage-IV cosmological experiment to go on-sky. The first cosmological analysis of the Y1 data acquisition was made public in Apr. 2024, and the Y3 data acquisition is done. I will present the DESI survey current status, along with the proposed extension of operations. I will then present the DESI-2 and Spec-S5 experiments. The DESI-2 experiment, currently planned for 2029-2035, will use the DESI instrument with no major upgrade and have three major programs: a 2 < z < 4.5 galaxy sample to constrain dark energy and primordial physics, a high-density z < 1 galaxy sample to constrain RSD and for cross-analyses with other experiments, and lastly a dwarf galaxies / stars sample to constrain dark matter. DESI-2 will be a pathfinder for the Spec-S5 experiment, which will be a major instrumental upgrade with respect to the DESI instrument.
