  In this paper, first we define Clairaut Riemannian map between Riemannian
manifolds by using a geodesic curve on the base space and find necessary and
sufficient conditions for a Riemannian map to be Clairaut with a non-trivial
example. We also obtain necessary and sufficient condition for a Clairaut
Riemannian map to be harmonic. Thereafter, we study Clairaut Riemannian map
from Riemannian manifold to Ricci soliton with a non-trivial example. We obtain
scalar curvatures of $rangeF_\ast$ and $(rangeF_\ast)^\bot$ by using Ricci
soliton. Further, we obtain necessary conditions for the leaves of
$rangeF_\ast$ to be almost Ricci soliton and Einstein. We also obtain necessary
condition for the vector field $\dot{\beta}$ to be conformal on $rangeF_\ast$
and necessary and sufficient condition for the vector field $\dot{\beta}$ to be
Killing on $(rangeF_\ast)^\bot$, where $\beta$ is a geodesic curve on the base
space of Clairaut Riemannian map. Also, we obtain necessary condition for the
mean curvature vector field of $rangeF_\ast$ to be constant. Finally, we
introduce Clairaut anti-invariant Riemannian map from Riemannian manifold to
K\"ahler manifold, and obtain necessary and sufficient condition for an
anti-invariant Riemannian map to be Clairaut with a non-trivial example.
Further, we find necessary condition for $rangeF_\ast$ to be minimal and
totally geodesic. We also obtain necessary and sufficient condition for
Clairaut anti-invariant Riemannian maps to be harmonic.
