We investigate departures from local thermodynamic equilibrium (LTE) in the line formation of Fe for a number of well-studied late-type stars in different evolutionary stages. A new model of the Fe atom was constructed from the most up-to-date theoretical and experimental atomic data available so far. Non-LTE (NLTE) line formation calculations for Fe were performed using 1D hydrostatic marcs and mafags-os model atmospheres, as well as the spatial and temporal average stratifications from full 3D hydrodynamical simulations of stellar convection computed using the stagger code. It is shown that the Fe i/Fe ii ionization balance can be well established with the 1D and mean 3D models under NLTE including calibrated inelastic collisions with H i calculated from Drawin's formulae. Strong low-excitation Fe i lines are very sensitive to the atmospheric structure; classical 1D models fail to provide consistent excitation balance, particularly so for cool metal-poor stars. A better agreement between Fe i lines spanning a range of excitation potentials is obtained with the mean 3D models. Mean NLTE metallicities determined for the standard stars using the 1D and mean 3D models are fully consistent. Moreover, the NLTE spectroscopic effective temperatures and gravities from ionization balance agree with that determined by other methods, e.g. the infrared flux method and parallaxes, if one of the stellar parameters is constrained independently.