  We study the shape, elasticity and fluctuations of the recently predicted
(cond-mat/9510172) and subsequently observed (in numerical simulations)
(cond-mat/9705059) tubule phase of anisotropic membranes, as well as the phase
transitions into and out of it. This novel phase lies between the previously
predicted flat and crumpled phases, both in temperature and in its physical
properties: it is crumpled in one direction, and extended in the other. Its
shape and elastic properties are characterized by a radius of gyration exponent
$\nu$ and an anisotropy exponent $z$. We derive scaling laws for the radius of
gyration $R_G(L_\perp,L_y)$ (i.e. the average thickness) of the tubule about a
spontaneously selected straight axis and for the tubule undulations
$h_{rms}(L_\perp,L_y)$ transverse to its average extension. For phantom (i.e.
non-self-avoiding) membranes, we predict $\nu=1/4$, $z=1/2$ and
$\eta_\kappa=0$, exactly, in excellent agreement with simulations. For
membranes embedded in the space of dimension $d<11$, self-avoidance greatly
swells the tubule and suppresses its wild transverse undulations, changing its
shape exponents $\nu$ and $z$. We give detailed scaling results for the shape
of the tubule of an arbitrary aspect ratio and compute a variety of correlation
functions, as well as the anomalous elasticity of the tubules. Finally we
present a scaling theory for the shape of the membrane and its specific heat
near the continuous transitions into and out of the tubule phase and perform
detailed renormalization group calculations for the crumpled-to-tubule
transition for phantom membranes.
