Brain diseases, including neurodegenerative diseases and neuropsychiatric diseases, have long plagued the lives of the affected populations and caused a huge burden on public health. Functional magnetic resonance imaging (fMRI) is an excellent neuroimaging technology for measuring brain activity, which provides new insight for clinicians to help diagnose brain diseases. In recent years, machine learning methods have displayed superior performance in diagnosing brain diseases compared to conventional methods, attracting great attention from researchers. This paper reviews the representative research of machine learning methods in brain disease diagnosis based on fMRI data in the recent three years, focusing on the most frequent four active brain disease studies, including Alzheimer's disease/mild cognitive impairment, autism spectrum disorders, schizophrenia, and Parkinson's disease. We summarize these 55 articles from multiple perspectives, including the effect of the size of subjects, extracted features, feature selection methods, classification models, validation methods, and corresponding accuracies. Finally, we analyze these articles and introduce future research directions to provide neuroimaging scientists and researchers in the interdisciplinary fields of computing and medicine with new ideas for AI-aided brain disease diagnosis.