An accurate abstractive summary of a document should contain all its salient information and should be logically entailed by the input document. We improve these important aspects of abstractive summarization via multi-task learning with the auxiliary tasks of question generation and entailment generation, where the former teaches the summarization model how to look for salient questioning-worthy details, and the latter teaches the model how to rewrite a summary which is a directed-logical subset of the input document. We also propose novel multi-task architectures with high-level (semantic) layer-specific sharing across multiple encoder and decoder layers of the three tasks, as well as soft-sharing mechanisms (and show performance ablations and analysis examples of each contribution). Overall, we achieve statistically significant improvements over the state-of-the-art on both the CNN/DailyMail and Gigaword datasets, as well as on the DUC-2002 transfer setup. We also present several quantitative and qualitative analysis studies of our model's learned saliency and entailment skills.