MODERN ALGEBRA based on the author's lectures on a one-semester course of abstract algebra at Fudan University, Shanghai, China is a concise introduction to algebraic structures up to basic Galois theory. The first four chapters cover the basic materials on groups, rings, fields, polynomials and linear algebra over arbitrary fields. The remaining four chapters include more advanced topics such as Sylow theorem, structure of finitely generated abelian groups, solvable groups, field extensions, finite fields and elementary Galois Theory. Ruler and compass construction and insolvability of quintic equations are included as applications of algebra. In the section on ring theory, the emphasis is on commutative rings and fields. The text is written for undergraduate students in math major who have taken a course of linear algebra. The brief solutions or hints of more than 150 exercises are provided for the convenience of the readers.