The development of autonomous vehicles has prompted an interest in exploring various techniques in navigation. One such technique is simultaneous localization and mapping (SLAM), which enables a vehicle to comprehend its surroundings, build a map of the environment in real time, and locate itself within that map. Although traditional techniques have been used to perform SLAM for a long time, recent advancements have seen the incorporation of neural network techniques into various stages of the SLAM pipeline. This review article provides a focused analysis of the recent developments in neural network techniques for SLAM-based localization of autonomous ground vehicles. In contrast to the previous review studies that covered general navigation and SLAM techniques, this paper specifically addresses the unique challenges and opportunities presented by the integration of neural networks in this context. Existing review studies have highlighted the limitations of conventional visual SLAM, and this article aims to explore the potential of deep learning methods. This article discusses the functions required for localization, and several neural network-based techniques proposed by researchers to carry out such functions. First, it presents a general background of the issue, the relevant review studies that have already been done, and the adopted methodology in this review. Then, it provides a thorough review of the findings regarding localization and odometry. Finally, it presents our analysis of the findings, open research questions in the field, and a conclusion. A semisystematic approach is used to carry out the review.