We study the role of Pt crystal orientation in spin-orbit torques in Co/Pt bilayers by means of the harmonic Hall effect and current-induced switching measurements. Perpendicularly magnetized Co/Pt bilayers were fabricated with the Pt layer exhibiting either a polycrystalline grain structure or an epitaxial (111)-oriented film on MgO substrates by magnetron sputtering. We find that the damping-like spin-orbit torque is 1.3 times smaller in the epitaxial Co/Pt(111) bilayers compared to the polycrystalline films, whereas the field-like spin-orbit torque values are of comparable magnitude. Current-induced magnetization switching measurements show good agreement with the results of harmonic measurements of damping-like torque, i.e., the critical switching current is about 30% higher in epitaxial Co/Pt(111). These results highlight the importance of crystal orientation effects on spin-orbit torques in nominally identical bilayer structures. Published under license by AIP Publishing.