This thesis considers the problem of planning a 3-dimensional rigid body motion with rotations. The task is associated with many practical usages: motion planning on assembly lines, device assembly checks and design of virtual prototypes. There exist two main classes of motion planning algorithms: exact and approximate. Both classes of algorithms can be characterized by different performance, quality of result and resource utilization. A survey of these algorithms leads to the conclusion that if an algorithm involves 3-dimensional rotations, there is always some kind of simplification in order to avoid motion planning in the space of rotations. The most common simplification is a discretization (splitting into small parts) of the space of rotations and considering only a translational motion in each fragment of space of rotations. To date, there is only a small number of research concerning motion in the 3-dimensional space of rotations. In this work, the author presents a study leading to the complete description of the configuration space of rotational movement in three dimensions and proposes three different algorithms for motion planning. The author presents his results and compares them with existing similar solutions. A few practical usages are also described. Among others, there is a possibility to use presented algorithms as procedures in a more general motion planning algorithm.