Many natural language questions require recognizing and reasoning with qualitative relationships (e.g., in science, economics, and medicine), but are challenging to answer with corpus-based methods. Qualitative modeling provides tools that support such reasoning, but the semantic parsing task of mapping questions into those models has formidable challenges. We present QUAREL, a dataset of diverse story questions involving qualitative relationships that characterize these challenges, and techniques that begin to address them. The dataset has 2771 questions relating 19 different types of quantities. For example, “Jenny observes that the robot vacuum cleaner moves slower on the living room carpet than on the bedroom carpet. Which carpet has more friction?” We contribute (1) a simple and flexible conceptual framework for representing these kinds of questions; (2) the QUAREL dataset, including logical forms, exemplifying the parsing challenges; and (3) two novel models for this task, built as extensions of type-constrained semantic parsing. The first of these models (called QUASP+) significantly outperforms off-the-shelf tools on QUAREL. The second (QUASP+ZERO) demonstrates zero-shot capability, i.e., the ability to handle new qualitative relationships without requiring additional training data, something not possible with previous models. This work thus makes inroads into answering complex, qualitative questions that require reasoning, and scaling to new relationships at low cost. The dataset and models are available at http://data.allenai.org/quarel.