Recently, numerous large language models (LLMs) have emerged as foundational models, reshaping biological data modeling and achieving remarkable breakthroughs in both discriminative and generative tasks. The success of these models is largely attributed to the inherent similarities between natural language and biological data, such as DNA, RNA, and amino acid sequences. Through pre-training and fine-tuning phases, LLMs have demonstrated their ability to effectively model these biological datasets. Additionally, while protein structures and RNA-seq expression data are not inherently sequential, they can still be modeled and predicted effectively by LLMs based on the Transformer architecture. Previous research has predominantly focused on architectural innovations in LLMs and their applications to sequential data across various domains. However, there is a notable lack of systematic reviews addressing the reasons and methods behind LLM modifications for fitting biological omics data, particularly for non-sequential data types. Furthermore, comprehensive analyses of LLM applications in synthetic biology remain limited. We first systematically review representative LLMs in the biological domain. Next, we delve into their applications across the genome, transcriptome, and proteome fields, detailing the goals, processes, datasets, and methodologies involved. Finally, we discuss the challenges of applying LLMs to biological omics data and fundamental scientific research. In summary, we aim to provide a comprehensive overview of the technical and conceptual advances in this field, as well as an essential resource for researchers exploring the diverse applications of LLMs across various biological disciplines.