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Abstractive text summarization aims at compressing the information of a long source document into a rephrased, condensed summary. Despite advances in modeling techniques, abstractive summarization models still suffer from several key challenges: (i) layout bias: they overfit to the style of training corpora; (ii) limited abstractiveness: they are optimized to copying n-grams from the source rather than generating novel abstractive summaries; (iii) lack of transparency: they are not interpretable. In this work, we propose a framework based on document-level structure induction for summarization to address these challenges. To this end, we propose incorporating latent and explicit dependencies across sentences in the source document into end-to-end single-document summarization models. Our framework complements standard encoder-decoder summarization models by augmenting them with rich structure-aware document representations based on implicitly learned (latent) structures and externally-derived linguistic (explicit) structures. We show that our summarization framework, trained on the CNN/DM dataset, improves the coverage of content in the source documents, generates more abstractive summaries by generating more novel n-grams, and incorporates interpretable sentence-level structures, while performing on par with standard baselines.