We use absolute trigonometric parallaxes from the Hipparcos Catalogue (Cat. <I/239>) to determine individual distances to members of the Hyades cluster, from which the 3-dimensional structure of the cluster can be derived. Inertially-referenced proper motions are used to rediscuss distance determinations based on convergent-point analyses. A combination of parallaxes and proper motions from Hipparcos, and radial velocities from ground-based observations, are used to determine the position and velocity components of candidate members with respect to the cluster centre, providing new information on cluster membership: 13 new candidate members within 20pc of the cluster centre have been identified. Farther from the cluster centre there is a gradual merging between certain cluster members and field stars, both spatially and kinematically. Within the cluster, the kinematical structure is fully consistent with parallel space motion of the component stars with an internal velocity dispersion of about 0.3km/s. The spatial structure and mass segregation are consistent with N-body simulation results, without the need to invoke expansion, contraction, rotation, or other significant perturbations of the cluster. The quality of the individual distance determinations permits the cluster zero-age main sequence to be accurately modelled. The helium abundance for the cluster is determined to be Y=0.26+/-0.02 which, combined with isochrone modelling including convective overshooting, yields a cluster age of 625+/-50Myr. The distance to the observed centre of mass (a concept meaningful only in the restricted context of the cluster members contained in the Hipparcos Catalogue) is 46.34+/-0.27pc, corresponding to a distance modulus m-M=3.33+/-0.01mag for the objects within 10pc of the cluster centre (roughly corresponding to the tidal radius). This distance modulus is close to, but significantly better determined than, that derived from recent high-precision radial velocity studies, somewhat larger than that indicated by recent ground-based trigonometric parallax determinations, and smaller than those found from recent studies of the cluster convergent point. These discrepancies are investigated and explained.