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 infalling galaxies. The cosmo-dynamical evolution of groups from initial Hubble expansion to collapse and virialization leads to a fundamental track (FT) in virial-theorem-M/L vs crossing time. The increased rates of mergers, both direct and after dynamical friction, in 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 somewhat less than the outermost radius where observed galaxy star formation efficiencies are enhanced relative 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 (lower) concentrations than LambdaCDM predictions in groups of high (low) mass. The critical mass, at M200 ~ 10^13 M_sun 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 FT both suggest that low velocity dispersion groups (compact and loose, X-ray emitting or undetected) are quite contaminated by chance projections.

Invited review, ESO workshop "Groups of Galaxies in the Nearby Universe", held in Santiago, Chile, 5-9 December 2005, ed. I. Saviane, V. Ivanov & J. Borissova, 16 pages