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Discourse coherence is strongly associated with text quality, making it important to natural language generation and understanding. Yet existing models of coherence focus on measuring individual aspects of coherence (lexical overlap, rhetorical structure, entity centering) in narrow domains. In this paper, we describe domain-independent neural models of discourse coherence that are capable of measuring multiple aspects of coherence in existing sentences and can maintain coherence while generating new sentences. We study both discriminative models that learn to distinguish coherent from incoherent discourse, and generative models that produce coherent text, including a novel neural latent-variable Markovian generative model that captures the latent discourse dependencies between sentences in a text. Our work achieves state-of-the-art performance on multiple coherence evaluations, and marks an initial step in generating coherent texts given discourse contexts.