  The electromagnetic radiation generated by ultra-relativistic positrons
channelling in a crystalline undulator is discussed. The crystalline undulator
is a crystal whose planes are bent periodically with the amplitude much larger
than the interplanar spacing. Various conditions and criteria to be fulfilled
for the crystalline undulator operation are established. Different methods of
the crystal bending are described. We present the results of numeric
calculations of spectral distributions of the spontaneous radiation emitted in
the crystalline undulator and discuss the possibility to create the stimulated
emission in such a system in analogy with the free electron laser. A careful
literature survey covering the formulation of all essential ideas in this field
is given. Our investigation shows that the proposed mechanism provides an
efficient source for high energy photons, which is worth to study
experimentally.
