Compact groups of galaxies have posed a number of challenging questions. Intensive observational and theoretical studies are now providing answers to many of these and, at the same time, are revealing unexpected new clues about the nature and role of these systems. Most compact groups contain a high fraction of galaxies having morphological or kinematical peculiarities, nuclear radio and infrared emission, and starburst or active galactic nuclei (AGN) activity. They contain large quantities of diffuse gas and are dynamically dominated by dark matter. They most likely form as subsystems within looser associations and evolve by gravitational processes. Strong galaxy interactions result and merging is expected to lead to the ultimate demise of the group. Compact groups are surprisingly numerous and may play a significant role in galaxy evolution.