Deep Neural Networks (DNNs) have emerged as a powerful and versatile set of techniques to address challenging artificial intelligence (AI) problems. Applications in domains such as image/video processing, natural language processing, speech synthesis and recognition, genomics and many others have embraced deep learning as the foundational technique. DNNs achieve superior accuracy for these applications using very large models which require 100s of MBs of data storage, ExaOps of computation and high bandwidth for data movement. Despite advances in computing systems, training state-of-the-art DNNs on large datasets takes several days/weeks, directly limiting the pace of innovation and adoption. In this paper, we discuss how these challenges can be addressed via approximate computing. Based on our earlier studies demonstrating that DNNs are resilient to numerical errors from approximate computing, we present techniques to reduce communication overhead of distributed deep learning training via adaptive residual gradient compression (AdaComp), and computation cost for deep learning inference via Prameterized clipping ACTivation (PACT) based network quantization. Experimental evaluation demonstrates order of magnitude savings in communication overhead for training and computational cost for inference while not compromising application accuracy.