The eXawatt Center for Extreme Light Studies project aimed to create a large scientific infrastructure based on lasers with giant peak power. The project relies on the significant progress achieved in the last decade. The planned infrastructure will incorporate a unique light source with a pulse power of 600 PW using optical parametric chirped pulse amplification in large-aperture KD2PO4, deuterated potassium dihydrogen phosphate crystals. The interaction of such laser radiation with matter represents a completely new fundamental physics. The direct study of the space–time structure of vacuums and other unknown phenomena at the frontier of high-energy physics and the physics of superstrong fields will be challenged. Expected applications will include the development of compact particle accelerators, the generation of ultrashort pulses of hard X-ray and gamma radiation for material science enabling one to probe material samples with unprecedented spatial and temporal resolution, the development of new radiation and particle sources, etc.