We describe Globus Genomics, a system that we have developed for rapid analysis of large quantities of next-generation sequencing (NGS) genomic data. This system achieves a high degree of end-to-end automation that encompasses every stage of data analysis including initial data retrieval from remote sequencing centers or storage (via the Globus file transfer system); specification, configuration, and reuse of multi-step processing pipelines (via the Galaxy workflow system); creation of custom Amazon Machine Images and on-demand resource acquisition via a specialized elastic provisioner (on Amazon EC2); and efficient scheduling of these pipelines over many processors (via the HTCondor scheduler). The system allows biomedical researchers to perform rapid analysis of large NGS datasets in a fully automated manner, without software installation or a need for any local computing infrastructure. We report performance and cost results for some representative workloads.