  The purpose of this paper is to organize some results on the local geometry
of CR singular real-analytic manifolds that are images of CR manifolds via a CR
map that is a diffeomorphism onto its image. We find a necessary (sufficient in
dimension 2) condition for the diffeomorphism to extend to a finite holomorphic
map. The multiplicity of this map is a biholomorphic invariant that is
precisely the Moser invariant of the image when it is a Bishop surface with
vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR
singular images and we prove that the set of CR singular points must be large,
and in the case of codimension 2, necessarily Levi-flat or complex. We also
show that there exist real-analytic CR functions on such images that satisfy
the tangential CR conditions at the singular points, yet fail to extend to
holomorphic functions in a neighborhood. We provide many examples to illustrate
the phenomena that arise.
