Sea-state estimation (SSE) supports safe, efficient, and autonomous maritime operation. Conventional sources, including wave buoys, satellites, radar systems, and metocean products, are valuable but cannot provide continuous, local estimates. Ship-motion-based SSE offers a complementary solution by using the vessel as a wave-sensing platform through the wave buoy analogy (WBA). This review examines machine-learning (ML) approaches to SSE from ship-motion responses published since 2018. The studies are synthesized by learning objective, data source, representation, and modeling strategy, covering regression, classification, directional wave-spectrum estimation, wave-elevation reconstruction, transfer learning, domain adaptation, and hybrid physics-ML methods. The synthesis shows that ML-based SSE has evolved from feasibility studies toward pipelines using simulated, experimental, and in-service data. Under controlled conditions, wave height and wave-period quantities are estimated more reliably than direction, spectral-shape parameters, secondary systems, and directional spectra. Full-scale studies demonstrate promise but reveal limitations from proxy labels, uneven coverage, missing vessel-state information, sensor heterogeneity, and temporal domain shift. Transfer learning and hybrid physics-ML can improve data efficiency and trustworthiness when domains are physically compatible and estimates are checked against WBA-based response consistency. Future priorities include richer operational datasets, deployment-oriented evaluation, uncertainty and explainability mechanisms, collaborative learning, multimodal representations, and foundation-model-inspired approaches grounded in vessel physics.