  We study systematically stationary solutions to the coupled Vlasov and
Poisson equations which have `self-similar' or scaling symmetry in phase space.
In particular, we find analytically {\it all} spherically symmetric
distribution functions where the mass density and gravitational potential are
strict power laws in $r$, the distance from the symmetry point. We treat as
special cases, systems built from purely radial orbits and systems that are
isotropic in velocity space. We then discuss systems with arbitrary velocity
space anisotropy finding a new and very general class of distribution
functions. These distributions may prove useful in modelling galaxies.
Distribution functions in cylindrical and planar geometries are also discussed.
Finally, we study spatially spheroidal systems that again exhibit strict
power-law behaviour for the density and potential and find results in agreement
with results published recently.
