  We present an algebraic framework which simultaneously generalizes the notion
of linear subspaces, matroids, valuated matroids, and oriented matroids. We
call the resulting objects matroids over hyperfields. In fact, there are (at
least) two natural notions of matroid in this context, which we call weak and
strong matroids. We give "cryptomorphic" axiom systems for such matroids in
terms of circuits, Grassmann-Plucker functions, and dual pairs, and establish
some basic duality theorems. We also show that if F is a doubly distributive
hyperfield then the notions of weak and strong matroid over F coincide.
