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Computational methods are an established tool in algebraic geometry and commutative algebra, the key element being the theory of Gröbner bases. This book represents the state of the art in computational algebraic geometry and encapsulates many of the most interesting trends and developments in the field. There are two articles on open problems, orienting the reader to the subject's direction, four surveys describing the most interesting work, and four original research papers. There is also an introduction to the theory of Gröbner bases and their use in computation. Though the perspective of the book is mathematical, it does relate the abstract and the experimental tendencies in the field. Consequently, it will appeal to computer scientists interested in symbolic computation, robotics or Gröbner bases, as well as mathematicians interested in algebraic geometry, commutative algebra, or the classification of algebras.