In this study, we define conformal hemi-slant Riemannian maps from an almost Hermitian manifold to a Riemannian manifold as a generalization of conformal anti-invariant Riemannian maps, conformal semi-invariant Riemannian maps and conformal slant Riemannian maps. Then, we obtain integrability conditions for certain distributions which are included in the notion of hemi-slant Riemannian maps and investigate their leaves. Also, we get totally geodesic conditions for this type maps. Lastly, we introduce some geometric properties under the notion of pluri-harmonic map.