An Introduction to Analysis on Metric Spaces offers a clear and accessible, yet rigorous treatment of the basic notions of Analysis, as well as the important theorems of the subjects. The subjects covered include, in the setting of metric spaces, convergence, continuity, completeness, compactness, connectedness, numerical series,  function series and Riemann integral. Then the author presents a comprehensive treatment of the big theorems of Analysis such as the Ascoli-Arzela Theorem, Stone Weierstrass Theorem and Baire Category Theorem. Written with students in mind, the book builds the essential foundations step by step, making it ideal for readers at the transition from elementary calculus to more advanced analysis. Each chapter is supported by carefully chosen examples and exercises that reinforce intuition while cultivating mathematical precision. This text serves as a solid starting point for anyone wishing to deepen their understanding of analysis in a general metric space setting students, instructors, and all enthusiasts of pure mathematics.