Metal thin films deposited on weakly-interacting substrates constitute an essential element of numerous microelectronic, catalytic, and optical devices. However, the natural tendency of metal atoms to agglomerate, upon condensation on a weakly-interacting surface, in dispersed three-dimensional (3D) islands affects negatively the performance of the above-mentioned devices. The aim of this thesis is to investigate one of the mechanisms governing silver (Ag) 3D island growth on weakly-interacting substrates, i.e. the nucleation of a new layer on the island top. Kinetic Monte Carlo (KMC) simulations are employed to calculate the top island-layer critical radius Rc required for nucleating a new layer in the out-of-plane direction. Single-island simulations are performed for growth temperatures T in the range 250 to 500 K and ratios of the pairwise adatom/substrate atom bond strength EB,sub to the corresponding adatom/adatom value EB,film in the range 0.5 to 0.75. We find that for T values below 250 K the islands exhibit a 2D morphology for all EB,sub/EB,film ratios. In contrast, for T values above 300 K there exists a range of relatively small EB,sub/EB,film values, where 2D morphology dominates. To calculate Rc for each island layer as the island shape evolves, a subroutine is developed and implemented in an existing KMC algorithm. Rc values are computed for 3D island growth at EB,sub/EB,film = 0.5 in the T range 300−500 K and the results show that Rc decreases monotonously from 17.3 to 6.0 Å and saturates approximately at 375 K. This trend is opposite to the typical behavior of islands grown under homoepitaxial conditions, for which the enhancement of downward inter-layer diffusion caused by an increase of T leads to lower atomic densities on the top, i.e. to a lower nucleation probability, and thus to an increase of Rc. This work contributes to the understanding of the physical processes that control thin-film morphological evolution; which is paramount for controlling and manipulating film growth for specific applications.