The promise of compressive sensing, exploitation of compressibility to achieve high quality image reconstructions with less data, has attracted a great deal of attention in the medical imaging community. At the Compressed Sensing Incubator meeting held in April 2014 at OSA Headquarters in Washington, DC, presentations were given summarizing some of the research efforts ongoing in compressive sensing for x-ray computed tomography and magnetic resonance imaging systems. This article provides an expanded version of these presentations. Sparsity-exploiting reconstruction algorithms that have gained popularity in the medical imaging community are studied, and examples of clinical applications that could benefit from compressive sensing ideas are provided. The current and potential future impact of compressive sensing on the medical imaging field is discussed.