There has been growing attention to multi-class classification problems, particularly those challenges of imbalanced class distributions. To address these challenges, various strategies, including data-level re-sampling treatment and ensemble methods, have been introduced to bolster the performance of predictive models and Artificial Intelligence (AI) algorithms in scenarios where excessive level of imbalance is present. While most research and algorithm development have been focused on binary classification problems, in health informatics there is an increased interest in the field to address the problem of multi-class classification in imbalanced datasets. Multi-class imbalance problems bring forth more complex challenges, as a delicate approach is required to generate synthetic data and simultaneously maintain the relationship between the multiple classes. The aim of this review paper is to examine over-sampling methods tailored for medical and other datasets with multi-class imbalance. Out of 2,076 peer-reviewed papers identified through searches, 197 eligible papers were chosen and thoroughly reviewed for inclusion, narrowing to 37 studies being selected for in-depth analysis. These studies are categorised into four categories: metric, adaptive, structure-based, and hybrid approaches. The most significant finding is the emerging trend toward hybrid resampling methods that combine the strengths of various techniques to effectively address the problem of imbalanced data. This paper provides an extensive analysis of each selected study, discusses their findings, and outlines directions for future research.