Computing the homotopy groups of spheres has been a central motivator in algebraic topology for decades. In this talk I’d like to unravel one approach to this problem, built around the idea of killing homotopy groups, and see how far it takes us. I’ll begin with the classical story of killing homotopy, as it appears in Serre’s finiteness theorem for the homotopy groups of spheres. Then I’ll turn this trick into an industrial-grade construction: the Adams-Novikov spectral sequence, a machine for computing stable homotopy groups. We will redeem this unraveling into a spectral sequence by an explicit description of the cobordism ring of complex manifolds! Through this identification I'll sketch a remarkable relationship between cobordism, cohomology, and formal group laws. This relationship forms the bedrock of chromatic homotopy theory, which organizes periodic phenomena in stable homotopy groups that I will briefly discuss at the end.