In the era of healthcare and its related research fields, the dimensionality problem of high-dimensional data is a massive challenge as it is crucial to identify significant genes while conducting research on diseases like cancer. As a result, studying new Machine Learning (ML) techniques for raw gene expression biomedical data is an important field of research. Disease detection, sample classification, and early disease prediction are all important analyses of high-dimensional biomedical data in the field of bioinformatics. Recently, machine-learning techniques have dramatically improved the analysis of high-dimension biomedical data sets. Nonetheless, researchers’ studies on biomedical data faced the challenge of vast dimensions, i.e., the vast features (genes) with a very low sample space. In this paper, two-dimensionality reduction methods, feature selection, and feature extraction are introduced with a systematic comparison of several dimension reduction techniques for the analysis of high-dimensional gene expression biomedical data. We presented a systematic review of some of the most popular nature-inspired algorithms and analyzed them. The paper is mainly focused on the original principles behind each of the algorithms and their applications for cancer classification and prediction from gene expression data. Lastly, the advantages and disadvantages of nature-inspired algorithms for biomedical data are evaluated. This review paper may guide researchers to choose the most effective algorithm for cancer classification and prediction for the satisfactory analysis of high-dimensional biomedical data.