  In this paper we give a formula for the $K$-theory of the $C^*$-algebra of a
weakly left-resolving labelled space. This is done by realising the
$C^*$-algebra of a weakly left-resolving labelled space as the Cuntz-Pimsner
algebra of a $C^*$-correspondence. As a corollary we get a gauge invariant
uniqueness theorem for the $C^*$-algebra of any weakly left-resolving labelled
space. In order to achieve this we must modify the definition of the
$C^*$-algebra of a weakly left-resolving labelled space.
  We also establish strong connections between the various classes of
$C^*$-algebras which are associated with shift spaces and labelled graph
algebras. Hence, by computing the $K$-theory of a labelled graph algebra we are
providing a common framework for computing the $K$-theory of graph algebras,
ultragraph algebras, Exel-Laca algebras, Matsumoto algebras and the
$C^*$-algebras of Carlsen.
  We provide an inductive limit approach for computing the $K$-groups of an
important class of labelled graph algebras, and give examples.
