  We classify, up to isomorphism and up to equivalence, division gradings (by
abelian groups) on finite-dimensional simple real algebras. Gradings on
finite-dimensional simple algebras are determined by division gradings, so our
results give the classification, up to isomorphism, of (not necessarily
division) gradings on such algebras.
  Linear algebra over the field of two elements plays an interesting role in
the proofs.
