In the first part of the book, written by Juergen Herzog, there is a survey on Stanley decompositions and their relation to the depth of a module, and discuss the conjecture by R. Stanley regarding this relationship. The conjecture remains wide open and the research around its solution has provided a collection of concepts and results. Professor Herzog has been very actively involved in this line of research and gives in this part a first hand survey of the topic. The tutorial of this part, written by Anna M. Bigatti and Emanuela De Negri uses the computer algebra system CoCoA-5 to compute Stanley decompositions and allows the reader to put his hands on the computational aspects of this topic.