Image captioning systems have recently improved dramatically, but they still tend to produce captions that are insensitive to the communicative goals that captions should meet. To address this, we propose Issue-Sensitive Image Captioning (ISIC). In ISIC, a captioning system is given a target image and an \emph{issue}, which is a set of images partitioned in a way that specifies what information is relevant. The goal of the captioner is to produce a caption that resolves this issue. To model this task, we use an extension of the Rational Speech Acts model of pragmatic language use. Our extension is built on top of state-of-the-art pretrained neural image captioners and explicitly reasons about issues in our sense. We establish experimentally that these models generate captions that are both highly descriptive and issue-sensitive, and we show how ISIC can complement and enrich the related task of Visual Question Answering.