This investigation is concerned with the theoretical determination of the behavior of the brain when the human head is subjected to torsional loadings. The mathematical model consists of a linear viscoelastic sphere which is bounded by and bonded to a rigid spherical shell. Two problems of torsional wave propagation from the rigid shell into the viscoelastic medium are solved in closed form. The first problem deals with dynamic response of the viscoelastic sphere when the angle of twist of the outer rigid shell about a vertical axis is a given function of time. The second problem is to determine the motion of the shell and viscoelastic sphere when the former is acted upon by a time dependent torque.