In the setting of multi-party computation, sets of two or more parties with private inputs wish to jointly compute some (predetermined) function of their inputs. General results concerning secure two-party or multi-party computation were first announced in the 1980s. Put briefly, these results assert that under certain assumptions one can construct prools for securely computing any desired multi-party functionality. However, this research relates only to a setting where a single prool execution is carried out. In contrast, in modern networks, many different prool executions are run at the same time. This book is devoted to the general and systematic study of secure multi-party computation under composition. Despite its emphasis on a theoretically well-founded treatment of the subject, general techniques for designing secure prools are developed that may even result in schemes or modules to be incorporated in practical systems. The book clarifies fundamental issues regarding security in a multi-execution environment and gives a comprehensive and unique treatment of the composition of secure multi-party prools.