  From a database of direct numerical simulations of homogeneous and isotropic
turbulence, generated in periodic boxes of various sizes, we extract the
spherically symmetric part of moments of velocity increments and first verify
the following (somewhat contested) results: the $4/5$-ths law holds in an
intermediate range of scales and that the second order exponent over the same
range of scales is {\it{anomalous}}, departing from the self-similar value of
$2/3$ and approaching a constant of $0.72$ at high Reynolds numbers. We compare
with some typical theories the dependence of longitudinal exponents as well as
their derivatives with respect to the moment order $n$, and estimate the most
probable value of the H\"older exponent. We demonstrate that the transverse
scaling exponents saturate for large $n$, and trace this trend to the presence
of large localized jumps in the signal. The saturation value of about $2$ at
the highest Reynolds number suggests, when interpreted in the spirit of
fractals, the presence of vortex sheets rather than more complex singularities.
In general, the scaling concept in hydrodynamic turbulence appears to be more
complex than even the multifractal description.
