Gas turbine thermal efficiency and fuel burn are very dependent on turbine entry temperature and overall pressure ratio (OPR). Unfortunately, increases in these two parameters compromise other key aspects of engine operation and tend to increase emissions of nitrogen oxides (NOx). The European Horizon 2020 ULTIMATE project researched advanced-cycle aero engines with synergistic combinations of novel technologies to increase thermal efficiency without increasing emissions. One candidate technology was the addition of secondary combustion to increase the mean temperature of heat addition to improve thermal efficiency while limiting the primary combustor flame temperatures and NOx formation. However, an overall reduction in NOx also requires the secondary combustor to be a low-NOx design. This paper describes numerical studies carried out on novel aero engine secondary combustor concepts developed in two MSc-thesis research projects. The studies have explored the potential of oxy-poor-flame combustion concepts. These annular combustor designs featured two distinct regions: (i) the vortex zone, which promotes recirculation of combustion products, a prerequisite for low-oxygen combustion, and (ii) a through-flow region where part of the incoming flow bypasses the vortex before the flows mix again. These studies have demonstrated the advantages and some limitations of the proposed designs and emissions assessments in comparison with previous secondary combustor studies. They suggest very low NOx is achievable with oxy-poor combustion, but will be more difficult if the incoming oxygen levels are above 10%. More-accurate assessments will require LES modelling and inclusion of the primary combustor in the simulations. However, if the low overall NOx emissions would include relatively higher levels of nitrous oxide (N2O) then this might raise concerns with respect to global warming.