A new family of fast O2- conductors, which exhibit anionic conductivity comparable to that of stabilised zirconia, is presented. The parent compound of this new family, hereafter called LAMOX, is lanthanum molybdate La2Mo2O9. Various substitutions have been attempted: on the lanthanum site (La(2-x)A(x))Mo2O9 with A = Sr, Ba, K, or Bi; on the molybdenum site La-2(Mo2-xBx)O-9 with B = Re, S, W, Cr and V; and on the oxygen site with fluorine. Most of these substitutions suppress the phase transition which occurs in La2Mo2O9 around 580 degreesC from a low temperature alpha form to a high temperature (more conducting) beta form, and stabilise the beta form at room temperature. Several members of the LAMOX series are studied through X-ray and neutron diffraction, and conductivity measurements. Large O2- thermal factors and local static disorder agree well with the anionic nature of the conductivity. Partly vacant sites with short inter-site distances suggest a most probable conduction path with tridimensional character.