The paper investigates the dissipation tensor $\unicode[STIX]{x1D700}_{ij}$ in wall turbulence. Available direct numerical simulation (DNS) data are examined to illustrate the differences in the anisotropy of the dissipation tensor $\unicode[STIX]{x1D700}_{ij}$ with respect to the anisotropy of the Reynolds stresses $\unicode[STIX]{x1D633}_{ij}$ . The budgets of the transport equations of the dissipation tensor $\unicode[STIX]{x1D700}_{ij}$ are studied using novel DNS data of low Reynolds number turbulent plane channel flow with spatial resolution sufficiently fine to accurately determine the correlations of products of two-derivatives of fluctuating velocities $u_{i}^{\prime }$ and pressure $p^{\prime }$ which appear in various terms. Finally, the influence of the Reynolds number on the diagonal components of $\unicode[STIX]{x1D700}_{ij}$ ( $\unicode[STIX]{x1D700}_{xx}$ , $\unicode[STIX]{x1D700}_{yy}$ , $\unicode[STIX]{x1D700}_{zz}$ ) and on the various terms in their transport equations is studied using available DNS data of Vreman and Kuerten ( Phys. Fluids , vol. 26, 2014 b , 085103).