Using first-principles calculations, we demonstrate that an Fe monolayer can assume very different magnetic phases on hcp ͑0001͒ and fcc ͑111͒ surfaces of 4d-and 5d-transition metals. Due to the substrates' d-band filling, the nearest-neighbor exchange coupling of Fe changes gradually from antiferromagnetic ͑AFM͒ for Fe films on Tc, Re, Ru, and Os to ferromagnetic on Rh, Ir, Pd, and Pt. In combination with the topological frustration on the triangular lattice of these surfaces the AFM coupling results in a 120°Néel structure for Fe on Re and Ru and an unexpected double-row-wise AFM structure on Rh, which is a superposition of left-and right-rotating 90°spin spirals. DOI: 10.1103/PhysRevB.79.094411 PACS number͑s͒: 75.70.Ak, 71.15.Mb Triggered by the discovery of the giant-magnetoresistance effect and the demand to realize spintronic device concepts, 1 magnetic nanostructures on surfaces have been a focus of experimental and theoretical research for more than 20 years now. In particular, there has been a tremendous effort to grow ultrathin transition-metal films on metal surfaces and to characterize and explain their magnetic properties. It is now generally believed that these structurally simple systems are well understood and more complex nanostructures such as atomic chains, clusters, or molecules on surfaces have moved into the spotlight of today's research. [2][3][4][5][6][7] Therefore, it came as a big surprise when it was experimentally shown that the prototypical ferromagnet Fe becomes a two-dimensional ͑2D͒ antiferromagnet on the W͑001͒ surface. 8 Combining spin-polarized scanning tunneling microscopy and first-principles calculations, it has been further demonstrated that complex magnetic order can be obtained even in single monolayer ͑ML͒ magnetic films on nonmagnetic substrates. For example, recently a spin-spiral state was discovered for a Mn ML on W͑110͒ ͑Ref. 9͒ and for a Mn ML on W͑001͒ ͑Ref. 10͒ and a nanoscale magnetic structure was found for an Fe ML on Ir͑111͒. 11 Surfaces of 4d-and 5d-transition metals ͑TMs͒ such as W, Re, Ru, or Ir have been particularly attractive from an experimental point of view as ultrathin 3d-TM films can often be grown pseudomorphically and without intermixing. 12-16 However, there has been controversy in the past about reports concerning dead magnetic layers and absence of magnetic order in ultrathin films on these surfaces. 14,15 The fundamental key to many unresolved puzzles may be the itinerant character of TMs resulting in competing exchange interactions beyond nearest neighbors and higher-order spin interactions beyond the Heisenberg model. The latter interactions have been proposed to play a role in transition metals; however, to our knowledge, no unambiguous proof of their importance has been given.Here, we use first-principles calculations to demonstrate that a hexagonal Fe ML can assume very different magnetic phases on a triangular lattice provided by hcp ͑0001͒ and fcc ͑111͒ surfaces of 4d-and 5d-transition metals, which are also experimentally accessib...