  We establish a noncommutative analogue of the first fundamental theorem of
classical invariant theory. For each quantum group associated with a classical
Lie algebra, we construct a noncommutative associative algebra whose underlying
vector space forms a module for the quantum group and whose algebraic structure
is preserved by the quantum group action. The subspace of invariants is shown
to form a subalgebra, which is finitely generated. We determine generators of
this subalgebra of invariants and determine their commutation relations. In
each case considered, the noncommutative modules we construct are flat
deformations of their classical commutative analogues. Thus by taking the limit
as $q\to 1$, our results imply the first fundamental theorem of classical
invariant theory, and therefore generalise them to the noncommutative case.
