Robotics research today is primarily about enabling autonomous, mobile robots to seamlessly interact with arbitrary, previously unknown environments. One of the most basic problems to be solved in this context is the question of where the robot is, and what the world around it, and in previously visited places looks like " the so-called simultaneous localization and mapping (SLAM) problem. We present a GraphSLAM system, which is a graph-based approach to this problem. This system consists of a frontend and a backend: The frontend- task is to incrementally construct a graph from the sensor data that models the spatial relationship between measurements. These measurements may be contradicting and therefore the graph is inconsistent in general. The backend is responsible for optimizing this graph, i. e. finding a configuration of the nodes that is least contradicting. The nodes represent poses, which do not form a regular vector space due to the contained rotations. We respect this fact by treating them as what they really are mathematically: manifolds. This leads to a very efficient and elegant optimization algorithm.