  We consider the Coulomb drag between two layers of two-dimensional electronic
gases subject to a strong magnetic field. We first focus on the case in which
the electronic density is such that the Landau level filling fraction $\nu$ in
each layer is at, or close to, $\nu=1/2$. Discussing the coupling between the
layers in purely electronic terms, we show that the unique dependence of the
longitudinal conductivity on wave-vector, observed in surface acoustic waves
experiments, leads to a very slow decay of density fluctuations. Consequently,
it has a crucial effect on the Coulomb drag, as manifested in the
transresistivity $\rho_D$. We find that the transresistivity is very large
compared to its typical values at zero magnetic field, and that its temperature
dependence is unique -- $\rho_D \propto T^{4/3}$. For filling factors at or
close to $1/4$ and $3/4$ the transresistivity has the same $T$-dependence, and
is larger than at $\nu = 1/2$. We calculate $\rho_D$ for the $\nu=3/2$ case and
propose that it might shed light on the spin polarization of electrons at
$\nu=3/2$. We compare our results to recent calculations of $\rho_D$ at
$\nu=1/2$ where a composite fermion approach was used and a
$T^{4/3}$-dependence was obtained. We conclude that what appears in the
composite fermion language to be drag induced by Chern-Simons interaction is,
physically, electronic Coulomb drag.
