This paper considers generalized method of moment–type estimators for which a criterion function is minimized that is not the “standard” quadratic distance measure but instead is a general Lp distance measure. It is shown that the resulting estimators are root-n consistent but not in general asymptotically normally distributed, and we derive the limit distribution of these estimators. In addition, we prove that it is not possible to obtain estimators that are more efficient than the “usual” L2-GMM estimators by considering Lp-GMM estimators. We also consider the issue of the choice of the weight matrix for Lp-GMM estimators.