A new mean-field theory of turbulent convection is developed by considering only the small-scale part of spectra as "turbulence" and the large-scale part, as a "mean flow," which includes both regular and semiorganized motions. The developed theory predicts the convective wind instability in a shear-free turbulent convection. This instability causes formation of large-scale semiorganized fluid motions in the form of cells. 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. This study predicts also the existence of the convective shear instability in a sheared turbulent convection. This instability causes formation of large-scale rolls and 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.