We have considered the problem of finding optimal transmission beams, or eigenmodes, for propagation through the random atmosphere with minimum distortion. We have found, making a realistic assumption that a transmitter knows atmospheric turbulence statistics, but not the instantaneous state of the atmosphere, that these stochastic eigenmodes are Laguerre-Gauss (LG) modes with beam waist chosen adequately relative to the field coherence length in the receiver plane. We have shown in our analysis that these eigenmodes have the remarkable capability of enduring coherence during propagation in turbulence. The ability to order the strength of individual stochastic eigenmodes notwithstanding turbulence is fundamental. It allows using of optimal transmission strategies that only consider the small number of most robust modes and significantly outperform conventional fixed mode sets under weak or strong turbulence conditions. Using these eigenmodes as transmit and receive bases minimizes signal degradation by turbulence, and minimizes the complexity of any signal processing for compensating the impact of turbulence. Adaptive optics can be replaced by adaptive multi-input multi-output signal processing, enabling compensation of fast fluctuations of both phase and amplitude.