We present the design and first performance and usability evaluation of GeMTC, a novel execution model and runtime system that enables accelerators to be programmed with many concurrent and independent tasks of potentially short or variable duration. With GeMTC, a broad class of such "many-task" applications can leverage the increasing number of accelerated and hybrid high-end computing systems. GeMTC overcomes the obstacles to using GPUs in a many-task manner by scheduling and launching independent tasks on hardware designed for SIMD-style vector processing. We demonstrate the use of a high-level MTC programming model (the Swift parallel dataflow language) to run tasks on many accelerators and thus provide a high-productivity programming model for the growing number of supercomputers that are accelerator-enabled. While still in an experimental stage, GeMTC can already support tasks of fine (subsecond) granularity and execute concurrent heterogeneous tasks on 86,000 independent GPU warps spanning 2.7M GPU threads on the Blue Waters supercomputer.