In the monograph we present some results of the work we have done in the theory of classical Cellular Automata (CA) and their appendices during 1969–2013, in truth with considerable pauses. These results at present form a rather essential constituent of the CA–problematics. In particular, we have studied such problems as the nonconstructability problem in CA, decomposition of global transition functions in CA, extremal constructive opportunities, the complexity of finite configurations and global transition functions, parallel formal grammars and languages defined by CA, the modelling problem in the classical CA, computer simulation of CA, certain applied aspects of CA, etc. At present, the CA–problematics is a rather well developed independent sphere of the mathematical cybernetics which has considerable field of numerous appendices. At that, with the equal right the CA–problematics can be considered as a component of such fields as discrete mathematics, the discrete parallel dynamic systems, complex systems and some others. In our opinion the book will present an indubitable interest for students, post–graduates and persons working for doctor's degree of the appropriate faculties of universities and colleges, first of all, of naturally scientific level along with teachers in such disciplines as cybernetics, automata theory, mathematics, mathematical and physical modelling, computer science, theoretical biology, computer technique, and a lot of others.