  Liquid crystal elastomers realize a fascinating new form of soft matter that
is a composite of a conventional crosslinked polymer gel (rubber) and a liquid
crystal. These {\em solid} liquid crystal amalgams, quite similarly to their
(conventional, fluid) liquid crystal counterparts, can spontaneously partially
break translational and/or orientational symmetries, accompanied by novel soft
Goldstone modes. As a consequence, these materials can exhibit unconventional
elasticity characterized by symmetry-enforced vanishing of some elastic moduli.
Thus, a proper description of such solids requires an essential modification of
the classical elasticity theory. In this work, we develop a {\em rotationally
invariant}, {\em nonlinear} theory of elasticity for the nematic phase of ideal
liquid crystal elastomers. We show that it is characterized by soft modes,
corresponding to a combination of long wavelength shear deformations of the
solid network and rotations of the nematic director field. We study thermal
fluctuations of these soft modes in the presence of network heterogeneities and
show that they lead to a large variety of anomalous elastic properties, such as
singular length-scale dependent shear elastic moduli, a divergent elastic
constant for splay distortion of the nematic director, long-scale
incompressibility, universal Poisson ratios and a nonlinear stress-strain
relation fo arbitrary small strains. These long-scale elastic properties are
{\em universal}, controlled by a nontrivial zero-temperature fixed point and
constitute a qualitative breakdown of the classical elasticity theory in
nematic elastomers. Thus, nematic elastomers realize a stable ``critical
phase'', characterized by universal power-law correlations, akin to a critical
point of a continuous phase transition, but extending over an entire phase.
