In this paper, as a generalization of pointwise slant submanifolds [B-Y. Chen and O. J. Garay, Pointwise slant submanifolds in almost Hermitian manifolds, Turk J Math 36, (2012), 630-640.], pointwise slant submersions [J.W.Lee and B. Şahin, Pointwise slant submersions, Bulletin of the Korean Mathematical Sosiety, 51(4), (2014), 115-1126.] and pointwise slant Riemannian maps [Y. Gündüzalp and M. A. Akyol, Pointwise slant Riemannian maps from Kaehler manifolds, Journal of Geometry and Physics, 179, (2002), 104589.], we introduce pointwise semi-slant Riemannian maps (briefly, PSSR maps) from almost Hermitian manifolds to Riemannian manifolds, present examples and characterizations. We also investigate the harmonicity of such maps. Moreover, we give Chen-Ricci inequality for a PSSR map. Finally, we study some curvature relations in complex space forms, involving Casorati curvatures for PSSR maps.