Data from e+e− annihilation into hadrons, collected by the JADE experiment at centre-of-mass energies between 14 GeV and 44 GeV, are used to study moments of event shape distributions. Models with hadronisation parameters tuned to the LEP 1 precision data provide an adequate description of the low energy data studied here. The NLO QCD calculations, however, show systematic deficiencies for some of the moments. The strong coupling measured from the moments which are reasonably described by NLO QCD, αS(MZ0)=0.1287±0.0007(stat.)±0.0011(exp.)±0.0022(had.)±0.0075(theo.), is consistent with the world average.