The effect of solvent on the collapse dynamics of polymers is studied using computer simulation. Two cases are investigated, one where the solvent is incorporated through a pairwise additive attraction between the polymer beads and a random force on each polymer bead, and another where the solvent is incorporated in an explicit fashion as a second component. Brownian dynamics and molecular dynamics simulations are used in the former and latter model, respectively, with intermolecular interactions chosen so that the equilibrium size of the polymer is similar in both models at similar conditions. In the Brownian dynamics simulations, at short times local blobs of monomers are found separated by linear segments. With time the blobs grow in size and coalesce to form sausage like shapes. These sausages gradually become thicker and shorter until the final shape of a spherical globule is reached. The first stage is rapid whereas the second sausage-sphere stage is slow. In this stage the polymer often gets trapped in local minima and the change in size with time occurs through discrete jumps, and the equilibrium conformation is often not reached. In contrast, in the molecular dynamics simulations with explicit solvent, the size of the polymer changes smoothly with time, and the polymer does not get trapped in local minima for the cases investigated, although the sequence of polymer shapes is similar. This suggests that incorporating solvent molecules explicitly is important in the computer simulations of collapse and folding of polymers.