Abstract: One of the simplest problems of normalization concerns analytic local diffeomorphisms of the plane in the neighborhood of a weakly attracting elliptic fixed point at which the derivative is a non-periodic rotation. Defining "geometric normalization" as an analytic local change of coordinates which transforms the local diffeomorphism into one leaving invariant the foliation by circles centered at the fixed point, I shall describe works with David Sauzin, Shanzhong Sun and Qiaoling Wei which address the existence of normalizations and geometric normalizations.