Torsion of an elastic half-space by a rigid punch is investigated. The boundary of the half-space is assumed to be rough. Two geometries of the punch-parabolic and flat end are considered. It is shown that the contact area consists of stick and slip zones. This fact, which is well-known in the classical torsional contact of the elastic half-space with the smooth surface and the parabolic punch, also holds true for the flat-ended punch if the boundary roughness is involved. The partial slip problems are reduced to the integral equations, which are solved numerically. The presented results show the effects of boundary roughness on the shear stresses, size of the stick area and the relation between the twisting moment and the angle of twist.