This book discusses various parts of the theory of mixed type partial differential equations with boundary conditions such as: Chaplygin''s classical dynamical equation of mixed type, the theory of regularity of solutions in the sense of Tricomi, Tricomi''s fundamental idea and one-dimensional singular integral equations on non-Carleman type, Gellerstedt''s characteristic problem and Frankl''s non-characteristic problem, Bitsadze and Lavrentjev''s mixed type boundary value problems, quasi-regularity of solutions in the classical sense. Some of the latest results of the author are also presented in this book.