Foreign Object Damage (FOD) occurs when small hard particles are ingested into aircraft jet engines. Particles impacting turbine blades at velocities up to about 300 m/s produce small indentation craters which can become sites for fatigue crack initiation, severely limiting the lifetime of the blade. FOD imparted to a thermal barrier system can cause delamination cracks extending away from the impact in the thermal barrier coating (TBC) adjacent to the interface. In this paper, a framework for analyzing the mechanics of FOD is established and its implication to fatigue cracking and TBC delamination is elaborated. Finite element analysis is used to determine the residual stresses and geometric stress concentration resulting from FOD. A non-dimensional analysis is presented that allows the impact and material variables to be grouped into the smallest possible parameter set needed to characterize FOD. This parameterization provides explicit results for the stresses and displacements that arise as the projectile characteristics and material properties are varied over a range applicable to FOD in gas turbines. The second step in the analysis focuses on the potency of cracks emerging from critical locations at the indents. The results have been used to address the question: When and to what extent do the residual stresses and stress concentration caused by FOD reduce the critical crack size associated with fatigue threshold? For deep indents, it is found that elastic stress concentration is the dominant factor when the applied cyclic load ratio is large, otherwise the residual stresses are also important. For FOD in a thermal barrier system, a scaling relation has been derived from the stress field and the penetration that relates the length of the interface delamination to the impact and material variables. Comparisons with a set of experiments conducted in parallel with the theory show that the numerical approach can account for various phenomena observed in practice.