the computation of stable homotopy groups of spheres is one of the most fundamental problems in topology. despite its simple definition, it is notoriously hard to compute. it has connections to many areas of mathematics. in this talk, i will discuss a recent breakthrough on this problem, which depends on motivic homotopy theory in a critical way. i will also talk about applications to smooth structures on spheres, and towards the open problem of kervaire invariant one in dimension 126. this talk is based on several joint work with bogdan gheorghe, daniel isaksen, and guozhen wang.