It is shown that each structure function $S_{n,m}(r)$, of order $n+m$, in strong turbulence is characterized by its own dissipation scale $\eta_{n,m}$. In the limit $n\rightarrow\infty$, the dissipation scale $\eta_{n,0}= O(\hbox{\it Re}^{-1})$, which is much smaller than the Kolmogorov scale $\eta=O(\hbox{\it Re}^{-{3}/{4}})$, $\hbox{\it Re}$ being the large-scale Reynolds number. This result has implications for the resolution requirements of direct simulations of turbulence. A new rigorous dynamic constraint relating scaling exponents of the structure functions to the codimension of the most singular features of turbulence is derived. A modification of the model by Gotoh & Nakano (2003) for the pressure–velocity correlations, based on the Bernoulli equation, is proposed. This proposal leads to an analytic expression for the scaling exponents of velocity structure functions.