This chapter provides a comprehensive yet practical introduction to the fundamentals of nonlinear seismic analysis—an essential framework for performance-based design and assessment of building structures. It begins by establishing the conceptual basis of structural nonlinearity, explaining how yielding, stiffness degradation, and energy dissipation govern inelastic response. Constitutive modeling of reinforced concrete and structural steel components is presented in detail, emphasizing the calibration of material and section behavior for reliable simulation. The chapter then discusses modeling considerations such as discretization, damping representation, second-order effects, and verification procedures to ensure numerical stability and physical realism. Two principal analysis methods are elaborated: the nonlinear static (pushover) analysis, which provides intuitive capacity–demand relationships and performance evaluation, and the nonlinear dynamic (response history) analysis, which directly integrates the inelastic equations of motion under ground motion time histories. Methods for verification, validation, and interpretation of results are addressed, followed by practical guidance on code applicability, documentation, and peer review. Finally, emerging trends—probabilistic and data-driven modeling, incremental dynamic analysis, and integration with digital twin and resilience-based design frameworks—are highlighted as key frontiers in the evolution of nonlinear seismic analysis. The chapter aims to bridge theory and engineering practice, equipping readers with both the conceptual understanding and procedural knowledge necessary to conduct credible nonlinear analysis for seismic performance evaluation.