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We present a new approach to dialogue management using conversational knowledge graphs as core representation of the dialogue state. To this end, we introduce a new dataset, GraphWOZ, which comprises Wizard-of-Oz dialogues in which human participants interact with a robot acting as a receptionist. In contrast to most existing work on dialogue management, GraphWOZ relies on a dialogue state explicitly represented as a dynamic knowledge graph instead of a fixed set of slots. This graph is composed of a varying number of entities (such as individuals, places, events, utterances and mentions) and relations between them (such as persons being part of a group or attending an event). The graph is then regularly updated on the basis of new observations and system actions. GraphWOZ is released along with detailed manual annotations related to the user intents, system responses, and reference relations occurring in both user and system turns. Based on GraphWOZ, we present experimental results for two dialogue management tasks, namely conversational entity linking and response ranking. For conversational entity linking, we show how to connect utterance mentions to their corresponding entity in the knowledge graph with a neural model relying on a combination of both string and graph-based features. Response ranking is then performed by summarizing the relevant content of the graph into a text, which is concatenated with the dialogue history and employed as input to score possible responses to a given dialogue state.