Today, computer games are one of the major sources of entertainment. Computer games are usually far more demanding than typical interactive applications in terms of motor and sensory skills needed for interaction control, due to special-purpose input devices, complicated interaction techniques, and the primary emphasis on visual control and attention. This renders computer games inaccessible to a large percentage of people with disabilities. This article introduces the concept of universally accessible games, that is, games proactively designed to optimally fit and adapt to individual gamer characteristics and to be concurrently played among people with diverse abilities, without requiring particular adjustments or modifications. The concept is elaborated and tested through four case studies: a web-based chess game (UA-Chess), an action game (Access Invaders), a universally inaccessible game (Game Over!) used as an interactive educational tool, and an improved version of Access Invaders (Terrestrial Invaders). For all cases, key design and evaluation findings are discussed, reporting consolidated know-how and experience. Finally, the research challenge of creating multiplayer universally accessible games is further elaborated, proposing the novel concept of Parallel Game Universes as a potential solution.