Alzheimer's disease (AD) is a neurodegenerative disorder that causes memory degradation and cognitive function impairment in elderly people. The irreversible and devastating cognitive decline brings large burdens on patients and society. So far, there is no effective treatment that can cure AD, but the process of early-stage AD can slow down. Early and accurate detection is critical for treatment. In recent years, deep-learning-based approaches have achieved great success in Alzheimer's disease diagnosis. The main objective of this paper is to review some popular conventional machine learning methods used for the classification and prediction of AD using Magnetic Resonance Imaging (MRI). The methods reviewed in this paper include support vector machine (SVM), random forest (RF), convolutional neural network (CNN), autoencoder, deep learning, and transformer. This paper also reviews pervasively used feature extractors and different types of input forms of convolutional neural network. At last, this review discusses challenges such as class imbalance and data leakage. It also discusses the trade-offs and suggestions about pre-processing techniques, deep learning, conventional machine learning methods, new techniques, and input type selection.