We survey recent progress in the use of analog memory devices to build neuromorphic hardware accelerators for deep learning applications. After an overview of deep learning and the application opportunities for deep neural network (DNN) hardware accelerators, we briefly discuss the research area of customized digital accelerators for deep learning. We discuss how the strengths and weaknesses of analog memory-based accelerators match well to the weaknesses and strengths of digital accelerators, and attempt to identify where the future hardware opportunities might be found. We survey the extensive but rapidly developing literature on what would be needed from an analog memory device to enable such a DNN accelerator, and summarize progress with various analog memory candidates including non-volatile memory such as resistive RAM, phase change memory, Li-ion-based devices, capacitor-based and other CMOS devices, as well as photonics-based devices and systems. After surveying how recent circuits and systems work, we conclude with a description of the next research steps that will be needed in order to move closer to the commercialization of viable analog-memory-based DNN hardware accelerators.