A low-profile luminaire has a planar light source comprised of one or more planar organic light emitting diodes (OLEDs). The OLED light source is supported by a low-profile OLED support structure that lies in a plane. The support structure has a maximum height (H) of about two inches and 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 of a grid ceiling system. Electrical components for driving and controlling the OLED light source can be provided within the support structure or externally of this structure.