This is an essential textbook the advanced undergraduate and graduate students of mathematics. Organized into seven chapters, the book treats real and p-adic groups in a unified manner. Chapter 1 outlines the preliminary material that will be used in the rest of the book. Chapter 2 is on analytic functions and is of an elementary nature. This material is included to cater to students who may not be familiar with p-adic fields. Chapter 3 introduces analytic manifolds and contains standard material. The only notable feature is that it covers both real and p-adic analytic manifolds. All the standard results on Lie groups are proved in Chaps. 4 and 5. Some of the proofs are, however, different from those in the earlier literature. Some results are not found in the literature, though they are kind of folklore among the experts in Lie theory. The last two chapters (Chaps. 6 and 7) are on Lie algebras and cover the structure theory as found in the first of the Bourbaki volumes on the subject. In these chapters, some proofs are new.