Phased arrays have a long history in radio astronomy. Large, sparse synthesis arrays have been in use for decades to capture high-resolution images of deep space objects. More recent work has extended the range of applications to other types of arrays, including aperture arrays (AAs) and phased array feeds (PAFs) for multibeam reflector antennas. The extreme sensitivity required for astronomical instrumentation is driving advances in numerical electromagnetic modeling, design optimization of large arrays, low noise amplifiers, minimization of receiver noise, cryogenic PAFs, array calibration, optimal beamforming, interferometric imaging, and array signal processing algorithms for radio-frequency interference mitigation. We give an overview of research progress, current and planned array-based instruments, and open challenges in these areas related to the new generation of sparse arrays, PAFs, and AAs that are in development for astronomical observatories around the world.