First Course in Metric Spaces presents a systematic and rigorous treatment of the subject of Metric Spaces which are mathematical objects equipped with the notion of distance. This book is a step towards the preparation for the study of more advanced topics in Analysis such as Topology. The contents are primarily suitable for teaching at the graduate level and serve as a treatise for undergraduate and postgraduate students. The book discusses classification of Metric Spaces using the standard classification notions such as completeness, compactness, and connectedness. The book also contains chapters on limit and continuity of mappings, fixed points and approximations, and a complete ordered field of real numbers obtained from cauchy sequences of rational numbers. Each chapter is interspersed with many examples and exercises.