Abstract: We use diffusion generative models to address two tasks of importance to cosmology -- as an emulator for cold dark matter density fields conditional on input cosmological parameters $\Omega_m$ and $\sigma_8$, and as a parameter inference model that can return constraints on the cosmological parameters of an input field. We show that the model is able to generate fields with power spectra that are consistent with those of the simulated target distribution, and capture the subtle effect of each parameter on modulations in the power spectrum. We additionally explore their utility as parameter inference models and find that we can obtain tight constraints on cosmological parameters.