We present smoothed particle hydrodynamics simulations of molecular cloud formation in spiral galaxies. These simulations model the response of a non-self-gravitating gaseous disc to a galactic potential. The spiral shock induces high densities in the gas, and considerable structure in the spiral arms, which we identify as molecular clouds. We regard the formation of these structures as due to the dynamics of clumpy shocks, which perturb the flow of gas through the spiral arms. In addition, the spiral shocks induce a large velocity dispersion in the spiral arms, comparable with the magnitude of the velocity dispersion observed in molecular clouds. We estimate the formation of molecular hydrogen, by post-processing our results and assuming the gas is isothermal. Provided the gas is cold (T≤ 100 K), the gas is compressed sufficiently in the spiral shock for molecular hydrogen formation to occur in the dense spiral arm clumps. These molecular clouds are largely confined to the spiral arms, since most molecular gas is photodissociated to atomic hydrogen upon leaving the arms.