We develop and analyze 1+1- and 2+1-dimensional (d) models for multilayer homoepitaxial growth of metal films at low temperatures (T), where intralayer terrace diffusion is inoperative. This work is motivated by recent variable-temperature scanning tunneling microscopy studies of Ag/Ag(100) homoepitaxy down to 50 K. Adsorption sites are bridge sites in our 1+1d models, and fourfold hollow sites in our 2+1d models for fcc(100) or bcc(100) surfaces. For growth at 0 K, we introduce a “restricted downward funneling” model, wherein deposited atoms can be trapped on the sides of steep nanoprotrusions rather than always funneling down to lower adsorption sites. This leads to the formation of overhangs and internal defects (or voids), and associated “rough” growth. Upon increasing T, we propose that a series of interlayer diffusion processes become operative, with activation barriers below that for terrace diffusion. This leads to “smooth” growth of the film for higherT (but still within the regime where terrace diffusion is absent), similar to that observed in models incorporating “complete downward funneling.”