Compact Lie group actions on contact manifolds The main result in this thesis is the classification of SO(3) actions on contact 5 manifolds. Using properties of the moment map one can reduce the manifold to a 3 dimensional contact manifold with an S^1 action. This works everywhere outside of the singular orbits. For the singular orbits three models can be given that describe all possible cases. The 5 manifold is then obtained by gluing the singular set onto the 3 dimensional S^1 manifold in a compatible way. As it is well known S^1 bundles over a closed surface are classified by an integer called the Euler number. A similar invariant can be recovered in our 3 dimensional setting. We call it the Dehn Euler number.