  We present a new method for extracting the true 3-d velocity and density
fields from the nonlinear redshift--space projected density field. The method
is based on the nonlinear, nonlocal transformation of the density field. We
assume a curl--free velocity field, although in the formulation presented here
this assumption can be dropped. Toy models with special symmetry are used to
test the intrinsic accuracy of the method compared to linear theory.
  Finding significant improvement we then test the method in more detail on
N--body simulations, considering the problems associated with sampling the
density field using sparse, magnitude--limited galaxy redshift surveys.
  A method of estimating the magnitude of shot--noise effects on the
deprojected velocity field is derived.
  We find there is a good correlation between the true and deprojected density
and velocity fields, even in the mildly nonlinear regime, provided that on
large scales galaxies trace the mass. An initial application is made to the
IRAS--QDOT all--sky redshift survey.
