**An Overview of Motivic Homotopy Theory **

* Marc Levine *

**Abstract**

Motivic homotopy theory was constructed by Morel and Voevodsky in the 1990s. It led to such striking applications as the solution of the Milnor conjecture and the Bloch-Kato conjecture on the Galois symbol. Since then the theory has turned to a systematic study of its basic invariants and structures, as well as providing numerous new applications. Morel, Cisinski-Déglise, and Röndigs-Østvær have described the relation of the motivic stable homotopy category to Voevodskys triangulated category of motives. Morel’s computation of the endomorphism of the sphere spectrum points out a close relationship to quadratic forms, and has given rise to the construction of interesting new oriented cycle theories. Asok and Fasel have applied computations of unstable motivic homotopy groups to stability problems for algebraic vector bundles. Isaksen and others have computed motivic versions of Adams-Novikov and Adams spectral sequences, and used this information to improve known computations of these spectral sequences in classical homotopy theory. We will discuss the basic ideas going into the construction of motivic homotopy theory and some of these results and applications.