Modern medicine is reliant on various medical imaging technologies for non-invasively observing patients’ anatomy. However, the interpretation of medical images can be highly subjective and dependent on the expertise of clinicians. Moreover, some potentially useful quantitative information in medical images, especially that which is not visible to the naked eye, is often ignored during clinical practice. In contrast, radiomics performs high-throughput feature extraction from medical images, which enables quantitative analysis of medical images and prediction of various clinical endpoints. Studies have reported that radiomics exhibits promising performance in diagnosis and predicting treatment responses and prognosis, demonstrating its potential to be a non-invasive auxiliary tool for personalized medicine. However, radiomics remains in a developmental phase as numerous technical challenges have yet to be solved, especially in feature engineering and statistical modeling. In this review, we introduce the current utility of radiomics by summarizing research on its application in the diagnosis, prognosis, and prediction of treatment responses in patients with cancer. We focus on machine learning approaches, for feature extraction and selection during feature engineering and for imbalanced datasets and multi-modality fusion during statistical modeling. Furthermore, we introduce the stability, reproducibility, and interpretability of features, and the generalizability and interpretability of models. Finally, we offer possible solutions to current challenges in radiomics research.