  We introduce affine Stanley symmetric functions for the special orthogonal
groups, a class of symmetric functions that model the cohomology of the affine
Grassmannian, continuing the work of Lam and Lam, Schilling, and Shimozono on
the special linear and symplectic groups, respectively. For the odd orthogonal
groups, a Hopf-algebra isomorphism is given, identifying (co)homology Schubert
classes with symmetric functions. For the even orthogonal groups, we conjecture
an approximate model of (co)homology via symmetric functions. In the process,
we develop type B and type D non-commutative k-Schur functions as elements of
the nilCoxeter algebra that model homology of the affine Grassmannian.
Additionally, Pieri rules for multiplication by special Schubert classes in
homology are given in both cases. Finally, we present a type-free
interpretation of Pieri factors, used in the definition of noncommutative
k-Schur functions or affine Stanley symmetric functions for any classical type.
