This chapter briefly reviews the present knowledge about the analytic information theory of quantum systems with non-standard dimensionality in the position and momentum spaces. The main concepts of this theory are the power and entropic moments, which are very fertile largely because of their flexibility and multiple interpretations. They are used here to study the most relevant information-theoretic one-element (Fisher, Shannon, Rényi, Tsallis) and some composite two-elements (Fisher-Shannon, LMC shape and Cramér-Rao complexities) measures which describe the spreading measures of the position and momentum probability densities farther beyond the standard deviation. We first apply them to general systems, then to single particle systems in central potentials and, finally, to hydrogenic systems in D-dimensions.