The technology of scintillating bolometers based on zinc molybdate (ZnMoO\(_4\)) crystals is under development within the LUMINEU project to search for 0\(\nu\)2\(\beta\) decay of \(^{100}\)Mo with the goal to set the basis for large scale experiments capable to explore the inverted hierarchy region of the neutrino mass pattern. Advanced ZnMoO\(_4\) crystal scintillators with mass of \(\sim\)~0.3 kg were developed and Zn\(^{100}\)MoO\(_4\) crystal from enriched \(^{100}\)Mo was produced for the first time by using the low-thermal-gradient Czochralski technique. One ZnMoO\(_4\) scintillator and two samples (59 g and 63 g) cut from the enriched boule were tested aboveground at milli-Kelvin temperature as scintillating bolometers showing a high detection performance. The first results of the low background measurements with three ZnMoO\(_4\) and two enriched detectors installed in the EDELWEISS set-up at the Modane Underground Laboratory (France) are presented.