A NATO Advanced Research Workshop "Computational Commutative and Non-Commutative Algebraic Geometry" was held in Chishinau, Moldova from 6 to 11 June, 2004. The financial support was offered by the NATO grant PST.ARW.980392. The workshop was organized by Gerhard Pfister and Svetlana Cojocaru with assistance of members of the Organizing Committee: Jürgen Herzog, Lorenzo Robbiano, Victor Ufnarovski. There were 25 lectures given by the participants and open discussions on the subjects presented. The central theme of this workshop was the interplay between commutative and noncommutative algebraic geometry, with its theoretical and computational aspects. The Scientific Program emphasizes current trends in Commutative and Non-Commutative Algebraic Geometry and Algebra. The contributors to this volume review the state of the art and present new evolution and progress reflecting the topics discussed in the lectures. The volume addresses in the first place researchers and graduate students. The editors would like to thank the contributors to this volume. Thanks are due to the NATO Public Diplomacy Division for their generous financial support. Special thanks to Tatiana Verlan, who had the task to correct and prepare the manuscript.

December 2004, Svetlana Cojocaru, Gerhard Pfister, Victor Ufnarovski