We use cosmological smoothed particle hydrodynamics (SPH) simulations of the Λ cold dark matter (CDM) model to study the abundance of damped Lyman α absorbers (DLAs) in the redshift range z= 0–4.5. We compute the cumulative DLA abundance by using the relation between the DLA cross-section and the total halo mass inferred from the simulations. Our approach includes standard radiative cooling and heating with a uniform ultraviolet background, star formation, supernova feedback and a phenomenological model for feedback by galactic winds. The latter allows us to examine, in particular, the effect of galactic outflows on the abundance of DLAs. We employ the ‘conservative entropy’ formulation of SPH developed by Springel & Hernquist, which mitigates against the systematic overcooling that affected earlier simulations. In addition, we utilize a series of simulations of varying box-size and particle number to isolate the impact of numerical resolution on our results. We show that the DLA abundance was overestimated in previous studies for three reasons: (i) the overcooling of gas occurring with non-conservative formulations of SPH, (ii) a lack of numerical resolution and (iii) an inadequate treatment of feedback. Our new results for the total neutral hydrogen mass density, DLA abundance and column density distribution function all agree reasonably well with observational estimates at redshift z= 3, indicating that DLAs arise naturally from radiatively cooled gas in dark matter haloes that form in a ΛCDM universe. Our simulations suggest a moderate decrease in DLA abundance by roughly a factor of 2 from z= 4.5 to 3, consistent with observations. A significant decline in abundance from z= 3 to 1, followed by weak evolution from z= 1 to 0, is also indicated, but our low-redshift results need to be interpreted with caution because they are based on coarser simulations than those employed at high redshift. Our highest resolution simulation also suggests that the halo mass-scale below which DLAs do not exist is slightly above 108h−1 M⊙ at z= 3–4, somewhat lower than previously estimated.