This book is concerned with the study of algebras of linear transformations acting on finite-dimensional vector spaces, which in one guise or another arise in various parts of modern mathematics. The book's aims are the following: (i) to give an exposition of the basic theory of finite-dimensional algebras at a level that is appropriate for senior undergraduate and first-year graduate students, and (ii) to provide the mathematical foundation that prepares the reader for the advanced study of any one of several fields of mathematics. The most important features of the book are the novelty of the selection of topics and the level of accessibility in which these topics are treated. The book features a relatively new approach to finite-dimensional operator algebras that is based on algebraic rather than functional-analytic methods. The reader who acquires a good understanding of the basic theory of algebras is well positioned to appreciate results in operator algebras, representation theory, linear algebra, and ring theory.