In this lecture I will review the earliest appearances of quadratic forms in algebraic topology, the index of a closed oriented manifold of dimension $4k$, and the Kervaire invariant as a subtle new invariant for dimensions $4k + 2$. The relation of this to surgery and groups of homotopy spheres will be discussed and some early definitions and constructions will be given, and the question of existence of Kervaire invariant 1 manifolds discussed. I will describe my definition of 1968 and how it led to the reduction of possibilities to dimensions of the form $2^q - 2$. This involves the notion of an exotic "orientation" of a special type , which exists for any manifold, but the choice of orientation can have significant consequences.