Algebraic geometry studies the geometry of sets of solutions of algebraic equations; it is a central part of modern mathematics. Although this vast and complicated world is governed by a rich conceptual framework, a lot of progress remains to be done to understand its innermost workings and applications for the advancement of modern mathematics and science in general. It has achieved its most spectacular developments in the study of Riemann surfaces, abelian varieties, and K3 surfaces. The EU-funded HyperK project aims to expand and generalize these advances to the world of hyperkähler geometry. The purpose is to verify in the hyperkähler world the general principles that govern modern algebraic geometry and prove fundamental results concerning algebraic cycles, Hodge structures, and cohomological invariants, thereby placing hyperkähler geometry at the centre of the field and enhancing our understanding of more general phenomena and applications.