This paper introduces a Unified Model of Saliency and Importance (UMSI), which learns to predict visual importance in input graphic designs, and saliency in natural images, along with a new dataset and applications. Previous methods for predicting saliency or visual importance are trained individually on specialized datasets, making them limited in application and leading to poor generalization on novel image classes, while requiring a user to know which model to apply to which input. UMSI is a deep learning-based model simultaneously trained on images from different design classes, including posters, infographics, mobile UIs, as well as natural images, and includes an automatic classification module to classify the input. This allows the model to work more effectively without requiring a user to label the input. We also introduce Imp1k, a new dataset of designs annotated with importance information. We demonstrate two new design interfaces that use importance prediction, including a tool for adjusting the relative importance of design elements, and a tool for reflowing designs to new aspect ratios while preserving visual importance.