AbstractDeep Contextual Language Models (LMs) like ELMO, BERT, and their successors dominate the landscape of Natural Language Processing due to their ability to scale across multiple tasks rapidly by pre-training a single model, followed by task-specific fine-tuning. Furthermore, multilingual versions of such models like XLM-R and mBERT have given promising results in zero-shot cross-lingual transfer, potentially enabling NLP applications in many under-served and under-resourced languages. Due to this initial success, pre-trained models are being used as ‘Universal Language Models’ as the starting point across diverse tasks, domains, and languages. This work explores the notion of ‘Universality’ by identifying seven dimensions across which a universal model should be able to scale, that is, perform equally well or reasonably well, to be useful across diverse settings. We outline the current theoretical and empirical results that support model performance across these dimensions, along with extensions that may help address some of their current limitations. Through this survey, we lay the foundation for understanding the capabilities and limitations of massive contextual language models and help discern research gaps and directions for future work to make these LMs inclusive and fair to diverse applications, users, and linguistic phenomena.