Drug side effects have become paramount concerns in drug safety research, ranking as the fourth leading cause of mortality following cardiovascular diseases, cancer, and infectious diseases. Simultaneously, the widespread use of multiple prescription and over-the-counter medications by many patients in their daily lives has heightened the occurrence of side effects resulting from Drug-Drug Interactions (DDIs). Traditionally, assessments of drug side effects relied on resource-intensive and time-consuming laboratory experiments. However, recent advancements in bioinformatics and the rapid evolution of artificial intelligence technology have led to the accumulation of extensive biomedical data. Based on this foundation, researchers have developed diverse machine learning methods for discovering and detecting drug side effects. This paper provides a comprehensive overview of recent advancements in predicting drug side effects, encompassing the entire spectrum from biological data acquisition to the development of sophisticated machine learning models. The review commences by elucidating widely recognized datasets and Web servers relevant to the field of drug side effect prediction. Subsequently, The study delves into machine learning methods customized for binary, multi-class, and multi-label classification tasks associated with drug side effects. These methods are applied to a variety of representative computational models designed for identifying side effects induced by single drugs and DDIs. Finally, the review outlines the challenges encountered in predicting drug side effects using machine learning approaches and concludes by illuminating important future research directions in this dynamic field.