  We consider all 1/2 BPS excitations of $AdS \times S$ configurations in both
type IIB string theory and M-theory. In the dual field theories these
excitations are described by free fermions. Configurations which are dual to
arbitrary droplets of free fermions in phase space correspond to smooth
geometries with no horizons. In fact, the ten dimensional geometry contains a
special two dimensional plane which can be identified with the phase space of
the free fermion system. The topology of the resulting geometries depends only
on the topology of the collection of droplets on this plane. These solutions
also give a very explicit realization of the geometric transitions between
branes and fluxes. We also describe all 1/2 BPS excitations of plane wave
geometries. The problem of finding the explicit geometries is reduced to
solving a Laplace (or Toda) equation with simple boundary conditions. We
present a large class of explicit solutions. In addition, we are led to a
rather general class of $AdS_5$ compactifications of M-theory preserving ${\cal
N} =2$ superconformal symmetry. We also find smooth geometries that correspond
to various vacua of the maximally supersymmetric mass-deformed M2 brane theory.
Finally, we present a smooth 1/2 BPS solution of seven dimensional gauged
supergravity corresponding to a condensate of one of the charged scalars.
