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The mathematical classification of complex bursting oscillations in multiscale excitable systems, seen for example in physics and neuroscience, has been the subject of active enquiry since the early 1980s. This classification problem is fundamental as it also establishes analytical and numerical foundations for studying complex temporal behaviours in multiple timescale models. This manuscript begins by reviewing the seminal works of Rinzel and Izhikevich in classifying bursting patterns of excitable cell models. Moreover, we recall an alternative, yet complementary, mathematical classification approach by Golubitsky, which together with the Rinzel-Izhikevich proposals provide the state-of-the-art foundations to the classification problem. Unexpectedly, while keeping within the Rinzel-Izhikevich framework, we find novel cases of bursting mechanisms not considered before. Moving beyond the state-of-the-art, we identify novel bursting mechanisms that fall outside the current classifications. This leads us towards a new classification, which requires the analysis of both the fast and the slow subsystems of an underlying slow-fast model. This new classification allows the dynamical dissection of a larger class of bursters. To substantiate this, we add a new class of bursters with at least two slow variables, which we denote folded-node bursters, to convey the idea that the bursts are initiated or annihilated via a folded-node singularity. In fact, there are two main families of folded-node bursters, depending upon the phase of the bursting cycle during which folded-node dynamics occurs. If it occurs during the silent phase, we obtain the classical folded-node bursting (or simply folded-node bursting). If it occurs during the active phase, we have cyclic folded-node bursting. We classify both families and give examples of minimal systems displaying these novel types of bursting behaviour.