The structural and magnetic properties of ultrathin layers of Mn grown on Rh͑001͒ surfaces have been investigated using scanning tunneling microscopy ͑STM͒ and ab initio density-functional calculations. STM shows perfect wetting of the Rh substrate by the first Mn monolayer ͑ML͒ accompanied by a high mobility of the Mn atoms and only little film-substrate intermixing. By way of contrast, the second monolayer is unstable against formation of multilayer islands at 300 K. However, this is only a transient effect and growth approaches a layer-by-layer mode in thicker films. At 170 K growth proceeds layer-by-layer throughout. The structure of the films is face-centered tetragonal ͑fct͒ without any indication of lateral lattice mismatch up to 20 ML thickness. The apparent height differences between terraces of different local thickness are consistent with an average axial ratio close to unity ͑near fcc͒ for 2-3-ML-thick films that decreases continuously to Ϸ0.94 in thicker films. Detailed spin-polarized density-functional calculations have been performed for the bulk and the ͑001͒ surface of strained fct Mn, as well as for free-standing and supported Mn/Rh͑001͒. The results demonstrate that structure and stability of surface and thin films are strongly influenced by antiferromagnetic ͑AFM͒ ordering. In-plane c͑2 ϫ 2͒ AFM is found at the surface while bulk and the deeper layers prefer layered AFM. The calculations reproduce and explain the growth instability of 2 ML films and reproduce qualitatively the thickness-dependent variation in the film structure while the tetragonal distortion imposed by the size mismatch is overestimated because with all known exchange-correlation functionals the size ratio of bulk Rh and Mn crystals is overestimated. The calculations show that interface alloying is slightly exothermic but kinetically hindered because of a high activation energy for exchange processes.