Summary: In this thesis we are concerned with adaptive discontinuous finite element methods for systems of hyperbolic conservations laws. Ultimately, we are interested in dynamic model adaptation for dimensionally heterogeneous flows. The necessary tools to the construction of an adaptive higher-order scheme are developed in separate parts of this thesis. First, we discuss the stabilization of higher-order finite volume and Runge-Kutta discontinuous Galerkin methods. A stabilization strategy based on the constrained linear reconstruction of piecewise constant data is proposed in order to deal with highly nonconforming meshes. The adaptation of an approximate solution in \(h\)-, \(p\)- and \(hp\)-adaptive simulations on arbitrary meshes is discussed from a practical perspective in the second part of this thesis. In particular, we describe the generalization of data structures for implementing \(hp\)-adaptive discontinuous finite element methods in the finite element library Dune-Fem. Finally, the third part of this thesis introduces the discontinuous finite element methods of first- and higher-order for mixed-dimensional flows. We describe an adaptive procedure that automatically detects regions of full and lower-dimensional dynamics in a discontinuous finite element solution. The accuracy and improved efficiency of the adaptive scheme are established in a number of numerical experiments.