Properties of groups of galaxies depend sensitively on the algorithm for group selection, and even the most recent catalogs of groups built from redshift-space selection should suffer from projections and inclusion of infalling galaxies. The cosmo-dynamical evolution of groups from initial Hubble expansion to collapse and virialization leads to a fundamental track in virial-theorem estimated M/L vs crossing time. The increased rates of mergers, both direct and after orbital decay by dynamical friction, in (low velocity dispersion) groups relative to clusters, explain the higher fraction of elliptical galaxies at given local number density in X-ray selected groups, relative to clusters, even when the hierarchical evolution of groups is considered. Galaxies falling into groups and clusters should later travel outwards to typically 2 virial radii, which is close to but somewhat less than the outermost radius where galaxy star formation efficiencies are observed to be enhanced relative [GRAPHICS] to field galaxies of same morphological type. An ongoing analysis of the internal kinematics of X-ray selected groups suggests that the radial profiles of line of sight velocity dispersion are consistent with isotropic NFW distributions for the total mass density, with higher concentrations in massive groups than ACDM predictions and lower concentrations in low mass groups. The critical mass, at M-200 approximate to M-circle dot, is consistent with possible breaks in the X-ray luminosity-temperature and Fundamental Plane relations. The internal kinematics of groups indicate that the M - T relation of groups should agree with that extrapolated from clusters with no break at the group scale. The analyses of observed velocity dispersion profiles and of the fundamental track both suggest that low velocity dispersion groups (compact and loose, X-ray emitting or undetected) are quite contaminated by chance projections.