Recently, the use of bio-inspired learning techniques such as Hebbian learning and its closely-related Spike-Timing-Dependent Plasticity (STDP) variant 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 Sparse and Predictive Coding-based Hebbian neural network architectures and the related 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 the literature so far. To this end, this survey covers a number of recent works in the field of neuromorphic continual learning based on state-of-the-art Sparse and Predictive Coding technology; provides background theory to help interested researchers 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.