Modern genomics and artificial intelligence converge at a breakthrough point with the emergence of transformer-based large language models (LLMs) for genomic analysis. This review examines how transformer-based models have advanced genomic sequence analysis through novel tokenization strategies, self-attention mechanisms, and pre-training/fine-tuning paradigms. Unlike prior broad surveys, we focus on three sub-areas of genomics where transformer architectures have had a substantive impact: sequence-level analysis, population-level genetics, including genetic imputation and genome-wide association analysis, and biodiversity research, while addressing challenges like genome-scale computational demands, training biases, and interpreting attention in biological contexts. Looking forward, we outline how emerging foundation models, multi-modal integration, and biologically informed attention mechanisms could transform our understanding of genome function, providing a roadmap for leveraging genome language models to decode genome complexity and advance functional and population-level genomics.