Recently, the use bio-plausible learning techniques such as Hebbian and Spike-Timing-Dependent Plasticity (STDP) have drawn significant attention for the design of compute-efficient AI systems that can continuously learn on-line at the edge. A key differentiating factor regarding this emerging class of neuromorphic continual learning system lies in the fact that learning must be carried using a data stream received in its natural order, as opposed to conventional gradient-based offline training where a static training dataset is assumed available a priori and randomly shuffled to make the training set independent and identically distributed (i.i.d). In contrast, the emerging class of neuromorphic continual learning systems covered in this survey must learn to integrate new information on the fly in a non-i.i.d manner, which makes these systems subject to catastrophic forgetting. In order to build the next generation of neuromorphic AI systems that can continuously learn at the edge, a growing number of research groups are studying the use of bio-plausible Hebbian neural network architectures and Spiking Neural Networks (SNNs) equipped with STDP learning. However, since this research field is still emerging, there is a need for providing a holistic view of the different approaches proposed in literature so far. To this end, this survey covers a number of recent works in the field of neuromorphic continual learning; provides background theory to help interested researchers to quickly learn the key concepts; and discusses important future research questions in light of the different works covered in this paper. It is hoped that this survey will contribute towards future research in the field of neuromorphic continual learning.