  We carry out a systematic investigation on floating bodies in real space
forms. A new unifying approach not only allows us to treat the important
classical case of Euclidean space as well as the recent extension to the
Euclidean unit sphere, but also the new extension of floating bodies to
hyperbolic space.
  Our main result establishes a relation between the derivative of the volume
of the floating body and a certain surface area measure, which we called the
floating area. In the Euclidean setting the floating area coincides with the
well known affine surface area, a powerful tool in the affine geometry of
convex bodies.
