This paper provides a detailed study of turbulent statistics and scale-by-scale budgets in turbulent Rayleigh–Bénard convection. It aims at testing the applicability of Kolmogorov and Bolgiano theories in the case of turbulent convection and at improving the understanding of the underlying inertial-range scalings, for which a general agreement is still lacking. Particular emphasis is laid on anisotropic and inhomogeneous effects, which are often observed in turbulent convection between two differentially heated plates. For this purpose, the SO(3) decomposition of structure functions and a method of description of inhomogeneities are used to derive inhomogeneous and anisotropic generalizations of Kolmogorov and Yaglom equations applying to Rayleigh–Bénard convection, which can be extended easily to other types of anisotropic and/or inhomogeneous flows. The various contributions to these equations are computed in and off the central plane of a convection cell using data produced by a direct numerical simulation of turbulent Boussinesq convection at $\hbox{\it Ra}\,{=}\,10^6$ and $\hbox{\it Pr}\,{=}\,1$ with aspect ratio $A\,{=}\,5$. The analysis of the isotropic part of the Kolmogorov equation demonstrates that the shape of the third-order velocity structure function is significantly influenced by buoyancy forcing and large-scale inhomogeneities, while the isotropic part of the mixed third-order structure function $\langle(\Delta\theta)^2\Delta\vec{u}\rangle$ appearing in the Yaglom equation exhibits a clear scaling exponent 1 in a small range of scales. The magnitudes of the various low $\ell$ degree anisotropic components of the equations are also estimated and are shown to be comparable to their isotropic counterparts at moderate to large scales. The analysis of anisotropies notably reveals that computing reduced structure functions (structure functions computed at fixed depth for correlation vectors $\boldsymbol{r}$ lying in specific planes only) in order to reveal scaling exponents predicted by isotropic theories is misleading in the case of fully three-dimensional turbulence in the bulk of a convection cell, since such quantities involve linear combinations of different $\ell$ components which are not negligible in the flow. This observation also indicates that using single-point measurements together with the Taylor hypothesis in the particular direction of a mean flow to test the predictions of asymptotic dimensional isotropic theories of turbulence or to calculate intermittency corrections to these theories may lead to significant bias for mildly anisotropic three-dimensional flows. A qualitative analysis is finally used to show that the influence of buoyancy forcing at scales smaller than the Bolgiano scale is likely to remain important up to $\hbox{\it Ra}\,{=}\,10^9$, thus preventing Kolmogorov scalings from showing up in convective flows at lower Rayleigh numbers.