The purpose of this book is twofold: to present some basic ideas in commutative algebra and algebraic geometry and to introduce some interesting topics of current research, centered around Grobner bases, resultants, and syzygies. The design of the text in each chapter consists of two parts: the fundamentals and the applications, which make it suitable for courses of various lengths, levels, and topics based on the mathematical background of the students. The presentation of the material combines definitions and proofs with an emphasis on concrete examples. The book also illustrates the use of software such as Mathematics: and Singular. The applications portion of the chapter is suitable for a reader who knows a little commutative algebra and algebraic geometry already, and serves as a guide to some interesting research topics.