Structure, morphology, and magnetism of epitaxial FexCo1-x alloy ultrathin films grown on Cu(1 0 0) in ultrahigh vacuum were investigated by a multi-technique approach over the whole composition range up to thicknesses of nine atomic monolayers (ML). Combining the results of the different techniques it is found that the films grow at room temperature in a distorted FCC structure with random chemical order. The amount and sign of the distortion depend on thickness and composition. Below approximate to 60 to 70% Fe content the alloy films are smooth, and exhibit two different vertical interlayer distances. At higher Fe concentrations several superstructures are observed, which are attributed to regular structural rearrangements. The structural relaxation above 4 ML thickness known from purl Fe films is not observed in the FeCo alloy films. Instead a gradual structural change towards a BCC(1 1 0) structure with increasing thickness occurs at higher Fe concentrations, and is still not complete in films as thick as 9 ML. The interrelation between morphology and structure in Fe_xCo_1−x/Cu(100) is discussed in terms of composition-dependent lattice parameters and strain. An expansion of the films in the plane leads to a stabilization of the FCC structure, whereas a compression leads to corrugated films with a tendency towards structural transformations. The absence of the relaxed FCC structure found in pure Fe films is explained by Co atoms acting as defects in the FCC structure of the alloy films either in a geometrical sense and/or because of their different magnetic moment with respect to the Fe atoms in both ferromagnetic and antiferromagnetic configurations of FCC Fe.