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Summary form only given as follows. For an active media, that is spatially extended along two coordinates, including relativistic electron beams with sheet and annular geometry, as well laser media, the use of two-dimensional (2D) distributed feedback is beneficial for providing spatial coherence of the radiation and, thus, can be used to increase the total radiation power. Such 2D feedback can be realized in planar and co-axial 2D Bragg resonators having double-periodic corrugations of the metallic side walls (either similar modulation of the surface of wave-guiding dielectric film). High selectivity of 2D Bragg resonators of planar and coaxial geometry is demonstrated for large Fresnel parameters in the frame of coupled-wave model as well in direct 3D simulations. Formation of high-Q eigenmodes in the vicinity of the Bragg resonance frequency has been explained by peculiarities of the dispersion characteristics of normal waves. Results of the theoretical analysis are validated by data obtained in "cold" microwave measurements. For practical applications of 2D Bragg structures in free-electron masers and other powerful sources of coherent radiation with high selectivity, the spatial structure of the fundamental eigenmode has a key importance. In 2D Bragg structures this is distributed uniformly over all the volume of the structure and thus is beneficial for generation of coherent radiation from a large size active medium. Modeling of nonlinear dynamics of FEM with 2D distributed feedback also demonstrates the advantages of this novel feedback mechanism for production of spatial coherent radiation from large size electron beams. Simulation results are compared with recent experimental results, where narrow frequency radiation was obtained from Ka band co-axial and W band planar 2D Bragg FEM, which were realized at Strathclyde University and Budker Institute.