The growth and structure of single-crystalline Ni_xMn_100−x films on Cu(001) were studied for concentrations of χ ≥ 13 and thicknesses of 0-15 monolayer (ML). Medium energy electron diffraction (MEED) curves revealed a layer-by-layer growth mode at a substrate temperature of T=300 K for alloy films with 40<χ ≤ 100 and coverages up to 14 ML. A smooth transition to a three-dimensional growth is concluded from the decaying amplitude of the MEED oscillations as the Mn content increases. We find a characteristic c(2x2) superstructure by low-energy electron diffraction (LEED) for alloy compositions in the vicinity of the equiatomic region. A kinematic analysis of specular LEED intensity curves was employed to determine the average interlayer spacing d, yielding a decreasing d with Ni content from 1.90 Å for χ=13 to 1.73 Å for pure Ni on Cu(001). For equiatomic c(2x2) NiMn/Cu(001) we propose a bulklike L10 crystal structure, which is characterized by an in-plane orientation of the bulk c axis. We show that Co grows layer by layer and assumes a p(1x1) structure on equiatomic c(2x2) NiMn/Cu(001). Using Co/Cu(001) as a substrate for equiatomic NiMn leads to a non-layer-by-layer growth of the alloy film and a diffuse LEED pattern with weak c(2x2) spots. The investigation of such bilayer structures by magneto-optical Kerr effect measurements (MOKE) indicates the presence of an antiferromagnetic (AFM) order in Ni_xMn_100−x films with χ close to the equiatomic composition and thicknesses above 8 ML at 300 K, as concluded from the coercivity enhancement.