This book covers most of the kwn results on reducibility of polymials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are t covered here, but several theorems on reducibility of polymials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix). The book also treats other subjects like Ritt's theory of composition of polymials, and properties of the Mahler measure, and it concludes with a bibliography of over 300 items. This unique work will be a necessary resource for all number theorists and researchers in related fields.