  The aim of this paper is to introduce Clairaut conformal submersions between
Riemannian manifolds. First, we find necessary and sufficient conditions for
conformal submersions to be Clairaut conformal submersions. In particular, we
obtain Clairaut relation for geodesics on the total manifolds of conformal
submersions, and prove that Clairaut conformal submersions have constant
dilation along their fibers, which are totally umbilical, with mean curvature
being gradient of a function. Further, we calculate the scalar and Ricci
curvatures of the vertical distributions of the total manifolds. Moreover, we
find a necessary and sufficient condition for Clairaut conformal submersions to
be harmonic. For a Clairaut conformal submersion we find conformal changes of
the metric on its domain or image, that give a Clairaut Riemannian submersion,
a Clairaut conformal submersion with totally geodesic fibers, or a harmonic
Clairaut submersion. Finally, we give two non-trivial examples of Clairaut
conformal submersions to illustrate the theory and present a local model of
every Clairaut conformal submersion with integrable horizontal distribution.
