Nonuniform fast Fourier transforms dominate the computational cost in many applications including image reconstruction and signal processing. We thus present a general-purpose GPU-based CUDA library for type 1 (nonuniform to uniform) and type 2 (uniform to nonuniform) transforms in dimensions 2 and 3, in single or double precision. It achieves high performance for a given user-requested accuracy, regardless of the distribution of nonuniform points, via cache-aware point reordering, and load-balanced blocked spreading in shared memory. At low accuracies, this gives on-GPU throughputs around 109 nonuniform points per second, and (even including hostdevice transfer) is typically 4-10x faster than the latest parallel CPU code FINUFFT (at 28 threads). It is competitive with two established GPU codes, being up to 90x faster at high accuracy and/or type 1 clustered point distributions. Finally we demonstrate a 5-12x speedup versus CPU in an X-ray diffraction 3D iterative reconstruction task at 10(-12) accuracy, observing excellent multi-GPU weak scaling up to one rank per GPU.