  A concept -- quantum order -- is introduced to describe a new kind of orders
that generally appear in quantum states at zero temperature. Quantum orders
that characterize universality classes of quantum states (described by {\em
complex} ground state wave-functions) is much richer then classical orders that
characterize universality classes of finite temperature classical states
(described by {\em positive} probability distribution functions). The Landau's
theory for orders and phase transitions does not apply to quantum orders since
they cannot be described by broken symmetries and the associated order
parameters. We find projective representations of symmetry groups (which will
be called projective symmetry groups) can be used to characterize quantum
orders. With the help of quantum orders and the projective symmetry groups, we
construct hundreds of symmetric spin liquids, which have SU(2), U(1) or $Z_2$
gauge structures at low energies. Remarkably, some of the stable quantum phases
support gapless excitations even without any spontaneous symmetry breaking. We
propose that it is the quantum orders (instead of symmetries) that protect the
gapless excitations and make algebraic spin liquids and Fermi spin liquids
stable. Since high $T_c$ superconductors are likely to be described by a
gapless spin liquid, the quantum orders and their projective symmetry group
descriptions lay the foundation for spin liquid approach to high $T_c$
superconductors.
