The directed design and discovery of compounds with pre-determined properties is a long-standing challenge in materials research. We provide a perspective on progress toward achieving this goal using generative models for chemical compositions and crystal structures based on a set of powerful statistical techniques drawn from the artificial intelligence community. We introduce the central concepts underpinning generative models of crystalline materials. Coverage is provided of early implementations for inorganic crystals based on generative adversarial networks and variational autoencoders through to ongoing progress involving autoregressive and diffusion models. The influence of the choice of chemical representation and the generative architecture is discussed, along with metrics for quantifying the quality of the hypothetical compounds produced. While further developments are required to enable realistic predictions drawn from richer structure and property datasets, generative artificial intelligence is already proving to be complementary to traditional materials design strategies.