arXiv:2610.23809v1 Announce Type: new Abstract: Lyapunov exponents (LEs) are fundamental metrics from nonlinear dynamical systems theory that quantify the average exponential rate of divergence or convergence of nearby trajectories in phase space. Over the past three decades, LEs have emerged as powerful tools across a wide range of power system problems, including transient and voltage stability assessment, chaos detection and mitigation, short-term load and renewable energy forecasting, and fault diagnosis. This paper presents a comprehensive review of these applications, covering more than 165 publications spanning from the early demonstrations of chaos in simplified machine models (1990s) to the most recent data-driven and machine-learning-augmented approaches applied to inverter-based resource (IBR)-dominated grids (2024--2026). The review is organized into four principal application domains: (i) stability analysis and real-time monitoring, (ii) chaos characterization and control, (iii) forecasting and predictability analysis, and (iv) protection, fault diagnosis, and condition monitoring. For each domain, we trace the evolution of methods, compare model-based and data-driven LE estimation strategies, identify key benchmarks and validation gaps, and outline the most pressing open research problems.