Deep neural networks (DNNs) achieve best-known accuracies in many machine learning tasks involved in image, voice, and natural language processing and are being used in an ever-increasing range of applications. However, their algorithmic benefits are accompanied by extremely high computation and storage costs, sparking intense efforts in optimizing the design of computing platforms for DNNs. Today, graphics processing units (GPUs) and specialized digital CMOS accelerators represent the state-of-the-art in DNN hardware, with near-term efforts focusing on approximate computing through reduced precision. However, the ever-increasing complexities of DNNs and the data they process have fueled an active interest in alternative hardware fabrics that can deliver the next leap in efficiency. Resistive crossbars designed using emerging nonvolatile memory technologies have emerged as a promising candidate building block for future DNN hardware fabrics since they can natively execute massively parallel vector-matrix multiplications (the dominant compute kernel in DNNs) in the analog domain within the memory arrays. Leveraging in-memory computing and dense storage, resistive-crossbar-based systems cater to both the high computation and storage demands of complex DNNs and promise energy efficiency beyond current DNN accelerators by mitigating data transfer and memory bottlenecks. However, several design challenges need to be addressed to enable their adoption. For example, the overheads of peripheral circuits (analog-to-digital converters and digital-to-analog converters) and other components (scratchpad memories and on-chip interconnect) may significantly diminish the efficiency benefits at the system level. Additionally, the analog crossbar computations are intrinsically subject to noise due to a range of device- and circuit-level nonidealities, potentially leading to lower accuracy at the application level. In this article, we highlight the prospects for designing hardware accelerators for neural networks using resistive crossbars. We also underscore the key open challenges and some possible approaches to address them.