Vision transformers (ViTs) have emerged as the dominant architecture for person re-identification (Re-ID) - the task of matching individuals across spatially disjoint surveillance cameras in the face of substantial variation in viewpoint, lighting, occlusion, and clothing. This article provides a focused review of transformer-based Re-ID, covering the CNN-to-ViT transition, pure transformer backbones, efficiency-oriented approximations, occlusion-hardened variants, parameter-efficient foundation-model adaptations, spatio-temporal models for video Re-ID, and unified multimodal frameworks. The analysis is organised along five axes: architectural progression from convolutional to self-attention representations; design choices governing efficiency, robustness, and modality coverage; foundation-model adaptation strategies; quantitative performance across established benchmarks; and outstanding research challenges. Under standard controlled benchmarks, ViT methods achieve 95%+ rank-1 accuracy in supervised settings, rising to 97.6% with CLIP-guided unsupervised clustering. Unified transformer frameworks support visible, infrared, and natural-language query modalities concurrently. Significant limitations persist, however: accuracy on cloth-changing benchmarks stalls at 40-60%, cross-domain transfer incurs 15-25% degradation, adversarial patch perturbations can reduce performance to near zero, and on-device inference cost constrains practical deployment. The review synthesises these findings, delineating where transformers have substantively advanced Re-ID and where fundamental challenges remain unsolved.