In the past decade, there has been a surge in research examining the use of voice and speech analysis as a means of detecting neurodegenerative diseases such as Alzheimer's. Many studies have shown that certain acoustic features can be used to differentiate between normal aging and Alzheimer's disease, and speech analysis has been found to be a cost-effective method of detecting Alzheimer's dementia. The aim of this review is to analyze the various algorithms used in speech-based detection and classification of Alzheimer's disease. A literature survey was conducted using databases such as Web of Science, Google Scholar, and Science Direct, and articles published from January 2020 to the present were included based on keywords such as ``Alzheimer's detection'', "speech," and "natural language processing." The ADReSS, Pitt corpus, and CCC datasets are commonly used for the analysis of dementia from speech, and this review focuses on the various acoustic and linguistic feature engineering-based classification models drawn from 15 studies. Based on the findings of this study, it appears that a more accurate model for classifying Alzheimer's disease can be developed by considering both linguistic and acoustic data. The review suggests that speech signals can be a useful tool for detecting dementia and may serve as a reliable biomarker for efficiently identifying Alzheimer's disease.