We propose a discrimination-aware learning method to improve both the accuracy and fairness of biased face recognition algorithms. The most popular face recognition benchmarks assume a distribution of subjects without paying much attention to their demographic attributes. In this work, we perform a comprehensive discrimination-aware experimentation of deep learning-based face recognition. We also propose a notational framework for algorithmic discrimination with application to face biometrics. The experiments include three popular face recognition models and three public databases composed of 64,000 identities from different demographic groups characterized by sex and ethnicity. We experimentally show that learning processes based on the most used face databases have led to popular pre-trained deep face models that present evidence of strong algorithmic discrimination. Finally, we propose a discrimination-aware learning method, Sensitive Loss, based on the popular triplet loss function and a sensitive triplet generator. Our approach works as an add-on to pre-trained networks and is used to improve their performance in terms of average accuracy and fairness. The method shows results comparable to state-of-the-art de-biasing networks and represents a step forward to prevent discriminatory automatic systems.