Named entities – typically expressed via proper nouns – play a key role in Natural Language Processing, as their identification and comprehension are crucial in tasks such as Relation Extraction, Coreference Resolution and Question Answering, among others. Tasks like these also often entail dealing with concepts – typically represented by common nouns – which, however, have not received as much attention. Indeed, the potential of their identification and understanding remains underexplored, as does the benefit of a synergistic formulation with named entities. To fill this gap, we introduce Concept and Named Entity Recognition (CNER), a new unified task that handles concepts and entities mentioned in unstructured texts seamlessly. We put forward a comprehensive set of categories that can be used to model concepts and named entities jointly, and propose new approaches for the creation of CNER datasets. We evaluate the benefits of performing CNER as a unified task extensively, showing that a CNER model gains up to +5.4 and +8 macro F1 points when compared to specialized named entity and concept recognition systems, respectively. Finally, to encourage the development of CNER systems, we release our datasets and models at https://github.com/Babelscape/cner.