This thesis investigates the dynamical evolution of systems orbiting within deeper potentials. Initially we use a simple satellite-halo interaction to study the dynamical processes that act on orbiting systems and we compare these results to analytical theory. Deep images of the Centaurus cluster reveal a spectacular arc of diffuse light that stretches for over 100 kpc and yet is just a few kpc wide. We use numerical simulations to show that this feature can be produced by the tidal debris of a spiral galaxy that has been disrupted by the potential of one of the central cD galaxies of the cluster. The evolution of sub-halos is then studied in a cosmological context using high resolution N-body simulations of galactic mass halos that form in a cold dark matter (CDM) simulation. CDM halos form via a complex series of mergers, accretion events and violent relaxation. Halos are non-spherical, have steep singular density profiles and contain many thousands of surviving dark matter substructure clumps. This will lead to several unique signatures for experiments that aim to detect dark matter either indirectly, through particle annihilation, or directly in a laboratory. For the first time it is possible to construct maps of the gamma-ray sky that result from the annihilation of dark matter particles within simulated dark matter halo distributions.