A detailed dynamical analysis of the nearby rich Norma cluster (ACO 3627) is presented. From radial velocities of 296 cluster members, we find a mean velocity of 4871 +/- 54 km/s and a velocity dispersion of 925 km/s. The mean velocity of the E/S0 population (4979 +/- 85 km/s) is offset with respect to that of the S/Irr population (4812 +/- 70 km/s) by `Delta v = 164 km/s in the cluster rest frame. This offset increases towards the core of the cluster. The E/S0 population is free of any detectable substructure and appears relaxed. Its shape is clearly elongated with a position angle that is aligned along the dominant large-scale structures in this region, the so-called Norma wall. The central cD galaxy has a very large peculiar velocity of 561 km/s which is most probably related to an ongoing merger at the core of the cluster. The spiral/irregular galaxies reveal a large amount of substructure; two dynamically distinct subgroups within the overall spiral-population have been identified, located along the Norma wall elongation. The dynamical mass of the Norma cluster within its Abell radius is 1 - 1.1 x 10^15 h^-1_73 M_Sun. One of the cluster members, the spiral galaxy WKK 6176 which recently was observed to have a 70 kpc X-ray tail, reveals numerous striking low-brightness filaments pointing away from the cluster centre suggesting strong interaction with the intracluster medium.
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