A new mean-field theory of turbulent convection is developed by considering only the small-scales part of spectra as "turbulence" and the large-scales part, as a "mean flow", which includes both regular and semi-organized motions. This theory yields results which are in an agreement with some well known experimental findings which lacked theoretical explanation, e.g., (a) the turbulent heat conductivity increases with the decrease of shear and the turbulent Prandtl number decreases from 1/2 to 1/8; (b) in the presence of the mean shear, the horizontal heat flux is directed against the regular mean flow. The convective wind instability in a shear-free turbulent convection is found. This instability causes formation of large-scale semi-organized fluid motions in the form of cells or rolls (convective wind). Spatial characteristics of these motions, such as the minimum size of the growing perturbations and the size of perturbations with the maximum growth rate, are determined. A convective-shear instability in a sheared turbulent covection is also found. This instability causes generation of convective-shear waves which have a nonzero hydrodynamic helicity. Increase of shear promotes excitation of the convective-shear instability. Applications of the obtained results to the atmospheric turbulent convection and the laboratory experiments on turbulent convection are discussed.