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<p>We survey temperature patterns and heat transport in convective boundary layers (CBLs) from the perspective that these are emergent properties of far-from-equilibrium, complex dynamical systems. We use the term 'plumes' to denote the temperature patterns, in much the same way that the term 'eddies' is used to describe patterns of motion in turbulent flows. We introduce a two-temperature (2T) toy model to connect the scaling properties of temperature gradients, temperature variance and heat transport to the geometric properties of plumes. We then examine temperature (<i>T</i>) probability density functions and <i>w</i>-<i>T</i> joint probability density functions, <i>T</i> spectra and <i>wT</i> cospectra observed both within and above the surface friction layer. Here <i>w</i> is vertical velocity. We interpret these in terms of the properties of the plumes that give rise to them. We focus first on the self-similarity property of the plumes above the SFL, and then introduce new scaling results from within the SFL, which show that <i>T</i> spectra and <i>wT</i> cospectra are not self-similar with height at small heights <i>z/z<sub>s</sub></i> < 0.1, but increasingly display properties associated with random diffusion. The CBL similarity parameters defined by McNaughton et al (Non-linear Processes in Geophysics 14, 257–271, 2007) are used throughout. We conclude by contrasting our interpretation of the role of buoyancy in CBL flows with that of Richardson (proc. Roy. Soc. London A 87, 354–373, 1920), whose ideas inform the current interpretation of the statistical fluid mechanics model of boundary-layer flows.</p>