We have undertaken a multiwavelength project to study the properties of X-ray selected groups at intermediate redshifts (0.2 < z < 0.6). Mulchaey et al. (2006) presented the spectroscopic confirmation of nine groups. Jeltema et al. (2006) presented the X-ray properties of these groups based on observations taken with XMM-Newton. These observations confirm that these systems contain luminous, diffuse gas with luminosities and temperatures consistent with the correlations seen among groups and clusters at low redshift. We find indications of excess entropy in these systems over self-similar predictions out to large radii. We have obtained deeper spectroscopic follow-up of seven groups with the Keck and Gemini telescopes. Along with the HST imaging, we study the galaxy populations in these groups, and our main results are summarized below.