The theory of motives in algebraic geometry is a cohomology theory for algebraic varieties. It is distinguished by the property that it unifies the "Weil cohomology" theories, namely Betti cohomology, de Rham cohomology, $$\ell$$-adic étale cohomology, and $$p$$-adic crystalline cohomology. In this theory, varieties can be decomposed into combinations of more fundamental components, called motives.