Language technologies have made enormous progress, especially with the introduction of large language models (LLMs). On traditional tasks such as machine translation and sentiment analysis, these models perform at near-human level. These advances can, however, exacerbate a variety of issues that models have traditionally struggled with, such as bias, evaluation, and risks. In this position paper, we argue that many of these issues share a common core: a lack of awareness of the factors, context, and implications of the social environment in which NLP operates, which we call social awareness. While NLP is getting better at solving the formal linguistic aspects, limited progress has been made in adding the social awareness required for language applications to work in all situations for all users. Integrating social awareness into NLP models will make applications more natural, helpful, and safe, and will open up new possibilities. Thus we argue that substantial challenges remain for NLP to develop social awareness and that we are just at the beginning of a new era for the field.