In order to develop more efficient types of gears, further investigations into the theories of engagement is necessary. Up until now most of the research work on the theories of engagement has been carried out separately on different groups, and based on individual types of profiles. This book aims at developing some universal theories, which can not only be used for all types of gears, but can also be utilized in other fields such as sculptured surfaces. The book has four characteristics: the investigations are concentrated on mismatched tooth surfaces; all the problems are dealt with from a differential geometry point of view; most theories and algorithms are universal in application; and the algorithms are easy to follow and can be used in real situations. In the process of developing the algorithms, the authors have introduced some mathematical methods which are believed to be innovative with regard to the theories of engagement known so far. A theoretical treatment is presented throughout the book, supported by numerical examples and experiments. With the computer programmes listed at the end of the volume, any of the proposed methods can be easily utilized in practice. The book is intended for postgraduate students, lecturers, professors, or research staff in mechanical/manufacturing engineering, mathematics and R&D departments of research institutes and universities. It should also be useful for engineers working in the gear manufacturing sector of industry.