This collection, mostly review chapters, covers results in different aspects of nonsmooth analysis related to, connected with or inspired by quasidifferential calculus. Some applications to various problems of mechanics and mathematics are discussed; numerical algorithms are described and compared; open problems are presented and studied. The goal of the book is to provide up-to-date information concerning quasidifferentiability and related topics. The state of the art in quasidifferential calculus is examined and evaluated by experts, both researchers and users. Quasidifferentiable functions were introduced in 1979 and the 20th anniversary of this development provides a good occasion to appraise the impact, results and perspectives of the field.