Task-oriented conversational systems often use dialogue state tracking to represent the user's intentions, which involves filling in values of pre-defined slots. Many approaches have been proposed, often using task-specific architectures with special-purpose classifiers. Recently, good results have been obtained using more general architectures based on pretrained language models. Here, we introduce a new variation of the language modeling approach that uses schema-driven prompting to provide task-aware history encoding that is used for both categorical and non-categorical slots. We further improve performance by augmenting the prompting with schema descriptions, a naturally occurring source of in-domain knowledge. Our purely generative system achieves state-of-the-art performance on MultiWOZ 2.2 and achieves competitive performance on two other benchmarks: MultiWOZ 2.1 and M2M. The data and code will be available at https://github.com/chiahsuan156/DST-as-Prompting.