  The classical cycle class map for a smooth complex variety sends cycles in
the Chow ring to cycles in the singular cohomology ring. We study two cycle
class maps for smooth real varieties: the map from the I-cohomology ring to
singular cohomology induced by the signature, and a new cycle class map defined
on the Chow-Witt ring. For both maps, we establish basic compatibility results
like compatibility with pullbacks, pushforwards and cup products. As a first
application of these general results, we show that both cycle class maps are
isomorphisms for cellular varieties.
