This thesis presents the extension of the lattice Boltzmann method (LBM) to the solution of the Fokker-Planck equation with the FENE force law, on a single lattice for the use of modelling the flows of polymeric liquids. First implementation and the basic theory of the LBM is discussed including the derivation of the equilibrium function as a discretisation of the Maxwell-Boltzmann distribution function using Gauss-Hermite quadrature and the recovery of the Navier-Stokes equations from the LBE by use of multiscale analysis. A review of the extension of the LBM to multiphase flow is presented including colour models, pseudo-potential models and free energy models. Numerical results for a colour model have been given. Current viscoelastic lattice Boltzmann methods are discussed including results validating the approach by Onishi et al. in the cases of simple shear flow and start up shear flow. A LBM for the Fokker-Planck equation with the FENE force law is developed based on a new Gauss quadrature rule that has been derived. The validity of this method is confirmed for small We by comparison with results by Ammar and Singh et al. where it gives good agreement. A LBM for the Fokker-Planck equation is then coupled with a macroscopic solver for the solvent velocity to solve start-up plane Couette flow. This approach is validated by comparison with results by Leonenko and Phillips.