This presentation provides a structured overview of core concepts in mathematical logic and model theory, beginning with the syntax and semantics of first-order logic. It covers foundational results such as Gödel's completeness theorem, the compactness theorem, and the Löwenheim–Skolem theorem, illustrating how these principles enable the analysis of mathematical structures. The talk further explores categoricity, the Łoś–Vaught test, and Morley's theorem, before transitioning into finite model theory and descriptive complexity, where it highlights connections between logical expressibility and computational complexity classes, concluding with tools like Ehrenfeucht–Fraïssé games for analyzing definability and equivalence between structures.