Neurological disorders are marked by neurodegeneration, leading to impaired cognition, psychosis, and mood alterations. These symptoms are typically associated with functional changes in both emotional and cognitive processes, which are often correlated with anatomical variations in the brain. Hence, brain structural magnetic resonance imaging (MRI) data have become a critical focus in research, particularly for predictive modeling. The involvement of large MRI data consortia, such as the Alzheimer's Disease Neuroimaging Initiative (ADNI), has facilitated numerous MRI-based classification studies utilizing advanced artificial intelligence models. Among these, convolutional neural networks (CNNs) and non-convolutional artificial neural networks (NC-ANNs) have been prominently employed for brain image processing tasks. These deep learning models have shown significant promise in enhancing the predictive performance for the diagnosis of neurological disorders, with a particular emphasis on Alzheimer's disease (AD). This review aimed to provide a comprehensive summary of these deep learning studies, critically evaluating their methodologies and outcomes. By categorizing the studies into various sub-fields, we aimed to highlight the strengths and limitations of using MRI-based deep learning approaches for diagnosing brain disorders. Furthermore, we discussed the potential implications of these advancements in clinical practice, considering the challenges and future directions for improving diagnostic accuracy and patient outcomes. Through this detailed analysis, we seek to contribute to the ongoing efforts in harnessing AI for better understanding and management of AD.