  We report on recent progress in understanding the tubular phase of
self-avoiding anisotropic membranes. After an introduction to the problem, we
sketch the renormalization group arguments and symmetry considerations that
lead us to the most plausible fixed point structure of the model. We then
employ an epsilon-expansion about the upper critical dimension to extrapolate
to the physical interesting 3-dimensional case. The results are $\nu=0.62$ for
the Flory exponent and $\zeta=0.80$ for the roughness exponent. Finally we
comment on the importance that numerical tests may have to test these
predictions.
