Compound icons are prevalent on signs, webpages, and infographics, effectively conveying complex and abstract concepts, such as "no smoking" and "health insurance", with simple graphical representations. However, designing such icons requires experience and creativity, in order to efficiently navigate the semantics, space, and style features of icons. In this paper, we aim to automate the process of generating icons given compound concepts, to facilitate rapid compound icon creation and ideation. Informed by ethnographic interviews with professional icon designers, we have developed ICONATE, a novel system that automatically generates compound icons based on textual queries and allows users to explore and customize the generated icons. At the core of ICONATE is a computational pipeline that automatically finds commonly used icons for sub-concepts and arranges them according to inferred conventions. To enable the pipeline, we collected a new dataset, Compicon1k, consisting of 1000 compound icons annotated with semantic labels (i.e., concepts). Through user studies, we have demonstrated that our tool is able to automate or accelerate the compound icon design process for both novices and professionals.