  The group theoretical approach to the relativistic wave equations in the de
Sitter and Anti-de Sitter spaces for spin~0 and 1/2 massive particles is
considered. The invariant wave equations which determines the appropriate
irreducible representations are constructed. The explicit solutions of these
equations possessing a simplicity and physical transparency are obtained
without use of the separation of variables method. The connection with the
general-covariant approach to wave equations in a curved space is established.
It is shown that the Anti-de Sitter space is the solution of the Einstein-Dirac
equations. The geometrical meaning of the mass quantization in the Anti-de
Sitter space and of the particle creation in de Sitter space is shown.
