Generative language modelling has surged in popularity with the emergence of services such as ChatGPT and Google Gemini. While these models have demonstrated transformative potential in productivity and communication, they overwhelmingly cater to high-resource languages like English. This has amplified concerns over linguistic inequality in natural language processing (NLP). This paper presents the first systematic review focused specifically on strategies to address data scarcity in generative language modelling for low-resource languages (LRL). Drawing from 54 studies, we identify, categorise and evaluate technical approaches, including monolingual data augmentation, back-translation, multilingual training, and prompt engineering, across generative tasks. We also analyse trends in architecture choices, language family representation, and evaluation methods. Our findings highlight a strong reliance on transformer-based models, a concentration on a small subset of LRLs, and a lack of consistent evaluation across studies. We conclude with recommendations for extending these methods to a wider range of LRLs and outline open challenges in building equitable generative language systems. Ultimately, this review aims to support researchers and developers in building inclusive AI tools for underrepresented languages, a necessary step toward empowering LRL speakers and the preservation of linguistic diversity in a world increasingly shaped by large-scale language technologies.