Geometry and Harmonic Superspace: some recent progress

This contribution follows the talk, given by F. Delduc at the conference SQS'2011 in Dubna, Russia (July 18-23, 2011). To a considerable extent it is a summary of known facts about the links between geometry and extended supersymmetry in d=1 mechanics, with emphasis on the harmonic superspace method created in Dubna in the 80's. Some recent developments based on ref. [1] are also presented.