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The energy- and flux-budget (EFB) theory developed previously for atmospheric stably stratified turbulence is extended to the surface layer in atmospheric convective turbulence. This theory is based on budget equations for turbulent energies and fluxes in the Boussinesq approximation. In the lower part of the surface layer in the atmospheric convective boundary layer, the rate of turbulence production of the turbulent kinetic energy (TKE) caused by the surface shear is much larger than that caused by the buoyancy, which results in three-dimensional turbulence of very complex nature. In the upper part of the surface layer, the rate of turbulence production of TKE due to the shear is much smaller than that caused by the buoyancy, which causes unusual strongly anisotropic buoyancy-driven turbulence. Considering the applications of the obtained results to the atmospheric convective boundary-layer turbulence, the theoretical relationships potentially useful in modeling applications have been derived. The developed EFB theory allows us to obtain a smooth transition between a stably stratified turbulence to a convective turbulence. The EFB theory for the surface layer in a convective turbulence provides an analytical expression for the entire surface layer including the transition range between the lower and upper parts of the surface layer, and it allows us to determine the vertical profiles for all turbulent characteristics, including TKE, the intensity of turbulent potential temperature fluctuations, the vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature), the integral turbulence scale, the turbulence anisotropy, the turbulent Prandtl number, and the flux Richardson number. (C) 2022 Author(s).