A method for learning local affine-covariant regions is presented. We showthat maximizing geometric repeatability does not lead to local regions, a.k.afeatures,that are reliably matched and this necessitates descriptor-basedlearning. We explore factors that influence such learning and registration: theloss function, descriptor type, geometric parametrization and the trade-offbetween matchability and geometric accuracy and propose a novel hardnegative-constant loss function for learning of affine regions. The affineshape estimator -- AffNet -- trained with the hard negative-constant lossoutperforms the state-of-the-art in bag-of-words image retrieval and widebaseline stereo. The proposed training process does not require preciselygeometrically aligned patches.The source codes and trained weights areavailable at https://github.com/ducha-aiki/affnet