Abstract We address the issue of numerical convergence in cosmological smoothed particle hydrodynamics simulations using a suite of runs drawn from the eagle project. Our simulations adopt subgrid models that produce realistic galaxy populations at a fiducial mass and force resolution, but systematically vary the latter in order to study their impact on galaxy properties. We provide several analytic criteria that help guide the selection of gravitational softening for hydrodynamical simulations, and present results from runs that both adhere to and deviate from them. Unlike dark matter-only simulations, hydrodynamical simulations exhibit a strong sensitivity to gravitational softening, and care must be taken when selecting numerical parameters. Our results–which focus mainly on star formation histories, galaxy stellar mass functions and sizes–illuminate three main considerations. First, softening imposes a minimum resolved escape speed, vε, due to the binding energy between gas particles. Runs that adopt such small softening lengths that $v_ε {\,\, \gtrsim \,\,}10\, {\rm km s^{-1}}$ (the sound speed in ionised ∼104 K gas) suffer from reduced effects of photo-heating. Second, feedback from stars or active galactic nuclei may suffer from numerical over-cooling if the gravitational softening length is chosen below a critical value, εeFB. Third, we note that small softening lengths exacerbate the segregation of stars and dark matter particles in halo centres, often leading to the counter-intuitive result that galaxy sizes increase as softening is reduced. The structure of dark matter haloes in hydrodynamical runs respond to softening in a way that reflects the sensitivity of their galaxy populations to numerical parameters.