A low-profile luminaire has a light source comprised of one or more organic light emitting diodes (OLEDs), LEDs or other area light sources. The one or more area light sources are supported by a low-profile support structure that lies in a plane. The support structure, which supports the area light source substantially in the plane of the support structure, preferably has a maximum height of about two inches and more preferably a height of about one inch or less. It also has perimeter dimensions that allow the luminaire to fit within and to be supported by the T-bar grid openings of a grid ceiling system. Electrical components for driving and controlling the light source can be provided within the support structure or externally of this structure.