This book is based on a graduate computer science course entitled "Symbolic Computational Algebra" taught by the author at New York University. This book is meant for graduate students with training in theoretical computer science, who would like to do research in computational algebra or to understand the algorithms underlying currently available symbolic computational systems such as Mathematica, Maple or Axiom. The four main topics covered are Gröbner bases, characteristic sets, resultants and semialgebraic sets.