With the advent of low-power neuromorphic computing systems, new possibilities have emerged for deployment in various sectors, like healthcare and transport, that require intelligent autonomous applications. These applications require reliable low-power solutions for sequentially adapting to new relevant data without loss of learning. Neuromorphic systems are inherently inspired by biological neural networks that have the potential to offer an efficient solution toward the feat of continual learning. With increasing attention in this area, we present a first comprehensive review of state-of-the-art neuromorphic continual learning (NCL) paradigms. The significance of our study is multi-fold. We summarize the recent progress and propose a plausible roadmap for developing end-to-end NCL systems. We also attempt to identify the gap between research and the real-world deployment of NCL systems in multiple applications. We do so by assessing the recent contributions in neuromorphic continual learning at multiple levels—applications, algorithms, architectures, and hardware. We discuss the relevance of NCL systems and draw out application-specific requisites. We analyze the biological underpinnings that are used for acquiring high-level performance. At the hardware level, we assess the ability of the current neuromorphic platforms and emerging nano-device-based architectures to support these algorithms in the presence of several constraints. Further, we propose refinements to continual learning metrics for applying them to NCL systems. Finally, the review identifies gaps and possible solutions that are not yet focused upon for deploying application-specific NCL systems in real-life scenarios.