On semiconductor substrates, the states induced inside the band gap by an adsorbate layer or by an ultra-thin deposited metal film may host rather long-lived (hundreds of femtoseconds) excitations, but detailed knowledge of their atomic and electronic structure is required to analyze the processes governing their relaxation.</br> Moreover, ultrafast pump-probe laser spectroscopy experiments have made it possible to explore the state-specific relaxation dynamics of hot carriers at surfaces and interfaces. Our goal is to provide a system-specific theoretical analysis of these data.</br> As one example, quantum well states (QWS) in thin Pb films offer a suitable probing ground for studying the competition between the relaxation mechanisms of electron-electron and electron-phonon scattering.</br> By comparing ab initio theoretical results of the vibrational modes of the $\sqrt{3}\times\sqrt{3}$ reconstructed SIC phase of Pb on Si(111) to experimental Raman data we find that this reconstruction is a good starting point for investigation of multilayer Pb films on Si(111).</br> We performed first-principles density-functional (DFT) calculations addressing the electronic band structure of few-atomic-layer Pb films on Si(111). In addition, phonon spectra and matrix elements for electron-phonon coupling within deformation potential theory were obtained from the DFT calculations. This enables us to calculate state-specific rate constants for the electron-phonon scattering in particular QWS. The contribution of electronic processes (impact ionization) to the lifetime can be estimated from the imaginary part of the electronic self-energy calculated in the GW approximation. Moreover, by numerically solving the rate equations for the occupation numbers in a prototypical electronic band structure coupled to a phononic heat bath, we are able to follow the dissipation of the electronic excitation energy to the Pb lattice vibrations over long time. The time scales extracted from the simulations are compared to the experimental data from time-resolved pump-probe experiments.</br> In contrast to relaxation in metallic bands, some surface preparations give rise to an insulating surface with Mott-Hubbard bands inside the substrate band gap. For one such system, an adsorbate layer of Sn on Si(111) with coverage of 1/3, we calculated both the electronic and phononic band structures. The strongly correlated electronic ground state is approximated within our DFT approach by an antiferromagnetic state treated with the HSE hybrid functional. Again, the deformation potentials for electron-phonon scattering have been obtained from our calculations. Since the Mott-Hubbard gap is found to be much larger than typical phononic quanta, the relaxation of hot carriers into the ground state is possible only by a multi-phonon process. This gives rise to a very long lifetime of the excitations which we estimate from our calculations.