We report on recent progress in understanding the tubular phase of self-avo
iding anisotropic membranes. After an introduction to the problem, we sketc
h the renormalization group arguments and symmetry considerations that lead
us to the most plausible fixed point structure of the model. We then emplo
y an E-expansion about the upper critical dimension to extrapolate to the p
hysical interesting 3-dimensional case. The results are nu = 0.62 for the F
lory exponent and zeta = 0.80 for the roughness exponent. Finally we commen
t on the importance that numerical tests may have to test these predictions
.