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. The paper is translation from Russian [Kvantovaya Elektronika 53, 95 (2023)].