Resistive random-access memory (ReRAM)-based processing-in-memory (PIM) architectures have recently become a popular architectural choice for deep-learning applications. ReRAM-based architectures can accelerate inferencing and training of deep learning algorithms and are more energy efficient compared to traditional GPUs. However, these architectures have various limitations that affect the model accuracy and performance. Moreover, the choice of the deep-learning application also imposes new design challenges that must be addressed to achieve high performance. In this paper, we present the advantages and challenges associated with ReRAM-based PIM architectures by considering Convolutional Neural Networks (CNNs) and Graph Neural Networks (GNNs) as important application domains. We also outline methods that can be used to address these challenges.