A damping support structure for preventing vibrations of an apparatus table supporting an apparatus for manufacturing ultra-high precision devices such as semiconductors and printed circuit boards. The apparatus table is supported in suspension by support members, with its apparatus mounting surface disposed at a low level. The support members are supported by stationary members through vertically extendible and contractible first air springs. Horizontally extendible and contractible second air springs are disposed between the stationary members and the apparatus table. The first and second air springs serve to mitigate vertical and horizontal vibrations of the apparatus table, respectively. At the same time, in response to vibrations of the apparatus table detected by vibration sensors, air supply from air supply devices to inside spaces of the first and second air springs is controlled to apply controlling forces to the apparatus table through the first and second air springs for displacing the apparatus table relative to the stationary members and maintaining the apparatus table in an absolutely stationary state.