  As a generalization of anti-invariant Riemannian submersions, we introduce
anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian
manifolds. We give examples and investigate the geometry of foliations which
are arisen from the definition of an anti-Riemannian map. Then we give a
decomposition theorem for the source manifold of such maps. We also find
necessary and sufficient conditions for anti-invariant Riemannian maps to be
totally geodesic and show that every pluriharmonic Lagrangian Riemannian map,
which is a special anti-invariant Riemannian map, from a K\"{a}hler manifold to
a Riemannian manifold is totally geodesic.
