An alternative definition of topological quantum field theory in 2+1 dimensions is discussed. The fundamental objects in this approach are not gauge fields as in the usual approach, but non-local observables associated with graphs. The classical theory of graphs is defined by postulating a simple diagrammatic rule for computing the Poisson bracket of any two graphs. The theory is quantized by exhibiting a quantum deformation of the classical Poisson bracket algebra, which is realized as a commutator algebra on a Hilbert space of states. The wavefunctions in this graph representation'' approach are functionals on an appropriate set of graphs. This is in contrast to the usual connection representation'' approach in which the theory is defined in terms of a gauge field and the wavefunctions are functionals on the space of flat spatial connections modulo gauge transformations.