Cartesian cut cell meshes offer efficient meshing procedures for complex geometries at the cost of less control over mesh element shapes. Cell sizes in particular can become arbitrarily small, leading to the so called small cell problem, meaning a severe restriction on the time stepping size for an explicit time stepping scheme. The Domain of Dependence stabilization method attempts to circumvent this restriction by introducing additional penalty terms on top of a higher order discontinuous Galerkin scheme. We present recent advances of this method for linear and nonlinear hyperbolic conservation laws on two-dimensional cut cell meshes. For linear systems we can proof a L2-stability result for the semidiscretization in space. The ability to choose the time stepping size independent of small cut cells as well as the preservation of the accuracy of the underlying DG scheme will be demonstrated by numerical results for both linear and nonlinear equations.