Multi-domain dialogue state tracking (DST) is a critical component for conversational AI systems. The domain ontology (i.e., specification of domains, slots, and values) of a conversational AI system is generally incomplete, making the capability for DST models to generalize to new slots, values, and domains during inference imperative. In this paper, we propose to model multi-domain DST as a question answering problem, referred to as Dialogue State Tracking via Question Answering (DSTQA). Within DSTQA, each turn generates a question asking for the value of a (domain, slot) pair, thus making it naturally extensible to unseen domains, slots, and values. Additionally, we use a dynamically-evolving knowledge graph to explicitly learn relationships between (domain, slot) pairs. Our model has a 5.80% and 12.21% relative improvement over the current state-of-the-art model on MultiWOZ 2.0 and MultiWOZ 2.1 datasets, respectively. Additionally, our model consistently outperforms the state-of-the-art model in domain adaptation settings.