In relation to transversely isotropic media, this paper presents a detailed study of those aspects of the propagation of homogeneous plane elastic waves which are essential to a basic understanding of the behaviour of surface waves. It is first shown how the ordering of the speeds of plane waves provides, directly and simply, a means of classifying the chosen materials, with the class label specifying the broad structure of the slowness surface and the location of its singular points. An examination of the shape of the outer sheet of the slowness surface follows, providing inter alia a complete account of the incidence of the various types of transonic states. The discussion turns next to exceptional waves, that is homogeneous plane waves which leave free of traction some family of parallel planes. The subset of the plane waves possessing this property is determined, after which the subset of the exceptional waves serving as limiting waves for an exceptional transonic state is picked out. Exceptional transonic states occur only when the axis of material symmetry lies either in the reference plane or at right angles to the reference vector and these orientations of the axis are referred to as $\alpha$ and $\beta$ configurations respectively. The exceptional states are arranged in a threefold classification, one class consisting of a continuous set of $\alpha$ configurations and the others discrete $\beta$ configurations. The paper ends with calculations of the limiting speed of the transonic state for the totality of $\alpha$ and $\beta$ configurations.