Data from e+e- annihilation into hadrons collected by the JADE experiment at centre-of-mass energies between 14 GeV and 44 GeV were used to study moments of event shape distributions. The data were compared with Monte Carlo models and with predictions from QCD NLO order calculations. The strong coupling constant measured from the moments is alpha_S(M_Z) = 0.1286 +/- 0.0007 (stat) +/- 0.0011 (expt) +/- 0.0022 (had) +/- 0.0068 (theo), alpha_S(M_Z) = 0.1286 +/- 0.0072 (total error), consistent with the world average. However, systematic deficiencies in the QCD NLO order predictions are visible for some of the higher moments.
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