This paper presents a systematic investigation of fairness in large-scale language models (LMs). We highlight that, despite their superior performance on a variety of natural language processing (NLP) tasks, LMs can inherit and amplify social biases associated with sensitive attributes such as gender and race. Therefore, this paper comprehensively reviews various existing fairness concepts and proposes a new classification scheme based on transformer architectures, including encoder-only, decoder-only, and encoder-decoder LMs. Experimental examples and results are presented for each fairness definition, and future research and open issues are discussed to advance the field.