Voevodsky used the triangulted category of motives and motivic homotopy theory in an essential way in his proof of the Milnor conjecture and his contribution to the proof of the Bloch-Kato conjecture. After these stunning successes, the field has had a remarkable further development, leading to a better understanding of the relation of algebraic K-theory with motivic cohomology via Voevodsky’s slice tower, opening up new applications of the theory of quadratic forms through Morel’s identification of the Grothendieck-Witt ring with the endomorphism ring of the motivic sphere spectrum and acquiring powerful new tools through Ayoub’s construction of the Grothendieck six functor formalism for the motivic stable homotopy category. These and other technical advances have enabled computations of basic “structure constants” of motivic homotopy theory, such as the homotopy groups of motivic spheres in low degree, yielded new results on splitting of vector bundles on affine varieties, as well as giving striking applications to classical homotopy theory through motivic versions of the Adams and Adams-Novikov spectral sequences. Many new cohomology theories for algebraic varieties, such as algebraic cobordism and other flavors of cobordism, such as symplectic or special linear cobordism, have been analyzed and applied. The problem of recognizing when a space is an infinite loop space, a basic problem in classical homotopy theory, has found a remarkable solution in the motivic setting through the theory of framed cobordism.