The elastic wavefield generated by a point source of tractions acting on the surface of a transversely isotropic half–space is studied. The symmetry axis of the solid is oriented arbitrarily with respect to the surface of the half–space. First, the integral representation of the time–harmonic Green's tensor is given. Then the complete far–field asymptotic approximation of a quasi–longitudinal (qP) and two quasi–shear (qSH and qSV) waves is derived. The qP wave is described by the leading term of the ray series, since there is only one arrival of this wave. The qSH wave is treated similarly everywhere apart from the so–called kissing–point boundary layer, where the qSH and qSV wavefronts are tangentially close to each other. A special asymptotic formula is obtained for this case. The qSV sheet of the wave surface is allowed to have conical points and cuspidal edges. Thus, the far–field approximation of the qSV wave involves ray–asymptotic expressions while inside the geometrical regions (where either one or three qSV arrivals exist), or else boundary–layer asymptotics inside conical–point, cuspidal–edge and kissing–point boundary layers. At the end of the paper we present numerical results of the simulation of pulse propagation. A good agreement between the asymptotic and direct numerical codes is achieved but the former is orders of magnitude faster.