This book arose out of courses the author taught in Abstract Algebra, Galois Theory, Algebraic Geometry, and Manifold Theory. It can be used for self-study or as a course textbook. It covers classic areas such as group theory, ring theory, linear algebra, and Galois theory. It also covers 20th century material on Grobner bases, category theory and cohomology. The section on category theory also covers some multilinear and exterior algebra.