  The Milnor conjecture has been a driving force in the theory of quadratic
forms over fields, guiding the development of the theory of cohomological
invariants, ushering in the theory of motivic cohomology, and touching on
questions ranging from sums of squares to the structure of absolute Galois
groups. Here, we survey some recent work on generalizations of the Milnor
conjecture to the context of schemes (mostly smooth varieties over fields of
characteristic not 2). Surprisingly, a version of the Milnor conjecture fails
to hold for certain smooth complete p-adic curves with no rational theta
characteristic (this is the work of Parimala, Scharlau, and Sridharan). We
explain how these examples fit into the larger context of an unramified Milnor
question, offer a new approach to the question, and discuss new results in the
case of curves over local fields and surfaces over finite fields.
