The treatment of homogeneous plane waves given in part I provides the basis for the detailed study of the nature of surface-wave propagation in transversely isotropic elastic media presented in this paper. The investigation is made within the framework of the existence theorem of Barnett and Lothe and the developments underlying its proof. The paper begins with a survey of this essential theoretical background, outlining in particular the formulation of the secular equation for surface waves in the real form F(v) = 0, F(v) being a nonlinear combination of definite integrals involving the acoustical tensor Q($\cdot$) and the associated tensor R($\cdot$,$\cdot$) introduced in part I. The calculation of F(v) for a transversely isotropic elastic material is next undertaken, first, in principle, for an arbitrary orientation of the axis of symmetry, then for the $\alpha$ and $\beta$ configurations, shown in part I to contain all the exceptional transonic states. In the rest of the paper the determination of F(v) is completed, in closed form, for the $\alpha$ and $\beta$ configurations and followed in each case by a discussion of the properties of F(v) and illustrative numerical results. This combination of analysis and computation affords a clear understanding of surface-wave behaviour in the exceptional configurations comprising, in the classification of part I, cases 1, 2 and 3. The findings for case 1 exhibit continuous transitions, within the $\alpha$ configurations, between subsonic and supersonic surface-wave propagation. Those for case 3 prove that there are discrete orientations of the axis for which no genuine surface wave can propagate and that this degeneracy typically has a marked influence on surface-wave properties in a sizeable sector of neighbouring $\beta$ configurations. Neither effect appears in previous accounts of surface-wave propagation in anisotropic elastic media.