Alzheimer's disease with progressive neurodegeneration and brain tumors notably characterized by rapid, not limited cell proliferation poses significant health risks unless timely diagnosed and treated. Tumors have a diverse feature and characteristics, added to subtle changes in the brain whose hallmark is Alzheimer's, making accurate segmentation and classification quite challenging. Indeed, while there have been research in the last decade or so that have proven promising results, challenges still linger on. The present work discusses various approaches for image classification and staging of Alzheimer's disease and brain tumors by exploiting techniques in statistical image processing and computational intelligence. This paper includes discussion on morphology of brain tumors along with neuroimaging changes caused by Alzheimer's disease, existing datasets, data augmentation techniques, and methods for component extraction and classification within the DL, TL, and ML framework. Such specific systems have been given the metrics using the datasets; the descriptions of the implementations, however may vary with the case at hand.