The state of the art of transient stability and steady-state (small signal) stability in power grids are reviewed. Transient stability concepts are illustrated with simple examples; in particular, we consider two machine learning-based methods for computing region of attraction: ROA produced by Neural Network Lyapunov Function; estimation of the ROA of IEEE 39-bus system using Gaussian process and Converse Lyapunov function. We discuss steady state stability in power systems, and using Prony’s modal analysis for evaluating small signal stability for the 7 Bus Test system and real French power system.