  Electric dipole moments constitute a competitive method to search for new
physics, being particularly sensitive to new CP-violating phases. Given the
experimental and theoretical progress in this field and more generally in
particle physics, the necessity for more reliable bounds than the ones usually
employed emerges. We therefore propose an improved extraction of the electric
dipole moment of the electron and the relevant coefficient of the
electron-nucleon coupling, taking into account theoretical uncertainties and
possible cancellations, to be used in model-dependent analyses. Specifically,
we obtain at 95% C.L. |d_e|<=0.14 10^{-26}e cm with present data, which is very
similar to the bound typically quoted from the YbF molecule, but obtained in a
more conservative manner. We examine furthermore in detail the prospects for
improvements and derive upper limits for the dipole moments of several
paramagnetic systems presently under investigation, i.e. Cesium, Rubidium and
Francium.
