Motivic homotopy theory is concerned with studying generalized cohomology theories for algebraic varieties (perhaps over characteristic p fields). I will explain a little about what this means, and discuss an application to a classical problem of Hurwitz concerning "sums-of-squares" identities. Certain techniques from this application lead one to think about quadratic bundles over schemes, which in particular can be used to construct an algebro-geometric analog of real K-theory (over any field!) I'll discuss some of this story.