Space debris poses a significant problem for the space sector. This problem relates to potential collisions of debris objects with active satellites, which in many cases will lead to catastrophic damage. Due to the absence of natural decay mechanisms in the higher regions of space, debris objects in these regions have very long orbital lifetimes. In order to assess the hazards posed to active satellites, it is relevant to be able to predict how the orbits of these debris objects behave on long timescales. A simulation code in C++ has been created for this thesis project, capable of efficient propagation of space debris trajectories over long periods of time (typically a century or more), while taking into account various relevant perturbing forces. The simulation code can be applied to simulate the orbits of debris objects with a wide range of area-to-mass ratios, from intact satellites to tiny flecks of paint. The results produced with the simulation code have been verified to be consistent with results presented in recent research papers on space debris. An extensive performance comparison has been made regarding the efficiency of different computational methods for carrying out accurate, long-term integrations of space debris orbits. Both traditional integration methods and symplectic integration methods were tested, the latter of which are interesting because of their energy conservation properties. All methods were also combined with different formulations of the equations of motion. Of the methods tested, the Dormand-Prince 8(7) integration method combined with Gauss' form of Lagrange's planetary equations in modified equinoctial elements was found to be the most efficient. The performance of the symplectic integration methods was markedly less for this application than for the integration of completely Hamiltonian systems, though it was certainly acceptable. The simulation code was also applied to predict the long-term orbital evolution for debris objects in GEO and GNSS graveyard orbits. While proposed GEO graveyard orbits were found to be safe, graveyard orbits in the GNSS region were found to be susceptible to resonances induced by the luni-solar perturbations, and hence, require a careful selection of the initial orbital parameters. In all cases, debris objects with high area-to-mass ratios were determined to be dangerous to active satellites, regardless of the initial conditions of the graveyard orbit.