The growing demand for physically motivated continuum theories of plasticity has led to an increased effort on dislocation based field descriptions. Only recently rigorous techniques have been developed by T. Hochrainer for performing meaningful averages over systems of moving, curved dislocations, which can be described by a higher order dislocation density tensor. Within this thesis we rewrite this continuum theory of dislocations using exclusively standard vector and tensor calculus. This formulation is much more accessible (although still defined in a higher order configuration space) than the original formulation which uses differential forms and higher order currents. This formulation then serves as the starting point for the numerical exploration of the continuum theory where we cover simple benchmark problems, which allow for verification with analytical solutions. This already demonstrates that within this theory it is possible to predict dislocation kinematics, which cannot be predicted by classical methods based e.g. on the 'Kröner-Nye tensor'. After this verification we then apply our numerical implementation to a complex example: bending of a thin film in a double slip configuration, which yields most interesting results concerning the general concept of 'geometrically necessary' and 'statistically stored' dislocations. Another most important outcome is that nearly all important kinematic properties of single dislocation lines are still contained and numerically accessible within this averaged continuum description.While we were pursuing the numerical exploration of the theory within this thesis, T. Hochrainer further developed his continuum theory towards a formulation which under certain simplifying assumptions does not require the higher order configuration space. This is extremely beneficial from point of view of computational cost and stability. A significant part of this thesis is concerned with verifying this simplified variant with the original formulation. The result is that in many physically relevant cases both theories yield very similar if not identical results.