Making autonomous driving possible requires extensive information about the surroundings as well as the state of the vehicle. While specific information can be obtained through singular sensors, a full estimation requires a multi sensory approach, including redundant sources of information to increase robustness. This thesis gives an overview of tasks that arise in sensor fusion in autonomous driving, and presents solutions at a high level of detail, including derivations and parameters where required to enable re-implementation. The thesis includes theoretical considerations of the approaches as well as practical evaluations. Evaluations are also included for approaches that did not prove to solve their tasks robustly. This follows the belief that both results further the state of the art by giving researchers ideas about suitable and unsuitable approaches, where otherwise the unsuitable approaches may be re-implemented multiple times with similar results. The thesis focuses on model-based methods, also referred to in the following as classical methods, with a special focus on probabilistic and evidential theories. Methods based on deep learning are explicitly not covered to maintain explainability and robustness which would otherwise strongly rely on the available training data. The main focus of the work lies in three main fields of autonomous driving: localization, which estimates the state of the ego-vehicle, mapping or obstacle detection, where drivable areas are identified, and object detection and tracking, which estimates the state of all surrounding traffic participants. All algorithms are designed with the requirements of autonomous driving in mind, with a focus on robustness, real-time capability and usability of the approaches in all potential scenarios that may arise in urban driving. In localization the state of the vehicle is determined. While traditionally global positioning systems such as a Global Navigation Satellite System (GNSS) are often used for this task, they are prone to errors and may produce jumps in the position estimate which may cause unexpected and dangerous behavior. The focus of research in this thesis is the development of a localization system which produces a smooth state estimate without any jumps. For this two localization approaches are developed and executed in parallel. One localization is performed without global information to avoid jumps. This however only provides odometry, which drifts over time and does not give global positioning. To provide this information the second localization includes GNSS information, thus providing a global estimate which is free of global drift. Additionally the use of LiDAR odometry for improving the localization accuracy is evaluated. For mapping the focus of this thesis is on providing a computationally efficient mapping system which is capable of being used in arbitrarily large areas with no predefined size. This is achieved by mapping only the direct environment of the vehicle, with older information in the map being discarded. This is motivated by the observation that the environment in autonomous driving is highly dynamic and must be mapped anew every time the vehicles sensors observe an area. The provided map gives subsequent algorithms information about areas where the vehicle can or cannot drive. For this an occupancy grid map is used, which discretizes the map into cells of a fixed size, with each cell estimating whether its corresponding space in the world is occupied. However the grid map is not created for the entire area which could potentially be visited, as this may be very large and potentially impossible to represent in the working memory. Instead the map is created only for a window around the vehicle, with the vehicle roughly in the center. A hierarchical map organization is used to allow efficient moving of the window as the vehicle moves through an area. For the hierarchical map different data structures are evaluated for their time and space complexity in order to find the most suitable implementation for the presented mapping approach. Finally for tracking a late-fusion approach to the multi-sensor fusion task of estimating states of all other traffic participants is presented. Object detections are obtained from LiDAR, camera and Radar sensors, with an additional source of information being obtained from vehicle-to-everything communication which is also fused in the late fusion. The late fusion is developed for easy extendability and with arbitrary object detection algorithms in mind. For the first evaluation it relies on black box object detections provided by the sensors. In the second part of the research in object tracking multiple algorithms for object detection on LiDAR data are evaluated for the use in the object tracking framework to ease the reliance on black box implementations. A focus is set on detecting objects from motion, where three different approaches are evaluated for motion estimation in LiDAR data: LiDAR optical flow, evidential dynamic mapping and normal distribution transforms. The thesis contains both theoretical contributions and practical implementation considerations for the presented approaches with a high degree of detail including all necessary derivations. All results are implemented and evaluated on an autonomous vehicle and real-world data. With the developed algorithms autonomous driving is realized for urban areas.