Results of numerical integrations are presented for a neutrally stratified planetary boundary layer containing a passive scalar, and for three unstable cases with upward heat flux. The air is assumed unsaturated. A total of either 16,000 or 32,000 grid points was used in a three-dimensional region with length and width several times the height of the boundary layer. A key result is the irrelevance of the neutral height scale, u*/f, and its replacement by the height zi of the inversion base which confines the convective mixing when &minuszi/L is as small as 1.5 (L is the Monin-Obukhoy length). Shapes of the eddies are examined for &minuszi/L=0, 1.5, 4.5 and 45; and only for the two slightly unstable cases were the vertical velocity eddies distinctly elongated as in Ekman-layer theories. At large instabilities it is shown how the friction velocity u/* loses its influence upon the turbulence intensifies and a convective velocity wale becomes important. Vertical profiles of mean wind, potential temperature, momentum flux, gross eddy coefficients, flux correlation coefficients, turbulence intensifies, temperature variance and pressure fluctuations are presented and compared, when possible, with measurements. Comparison is hindered by the lack of observations of zi and L in almost all field studies. Various terms in the turbulence kinetic energy equation, which are difficult to measure, are discussed quantitatively. The rate at which particles released near the surface are transported vertically by the calculated turbulence is found from Lagrangian integrations to be up to two orders of magnitude greater in unstable cases than in a typical neutral case.