$BiFeO_3$ films were hetero-epitaxially grown on $SrTiO_3$ substrate with a various orientation by pulse laser deposition. $BiFeO_3$ films grown on (111) $SrTiO_3$ substrate have a rhombohedral structure, identical to that of single crystals. On the other hand, films grown on (110) or (001) $SrTiO_3$ substrate are monoclinically distorted from the rhombohedral structure due to the epitaxial constraint. The easy axis of spontaneous polarization is close to [111] for the variously oriented films. Dramatically enhanced polarization and magnetization have been found for $BiFeO_3$ thin films grown on $SrTiO_3$ substrate comparing to that of $BiFeO_3$ crystals. The results are explained in terms of an epitaxially-induced transition between cycloidal and homogeneous spin states, via magneto-electric interactions.