Abstract: Alloys of silicon (Si), germanium (Ge) and tin (Sn) are continuously attracting research attention as possible direct band gap semiconductors with prospective applications in optoelectronics. The direct gap property may be brought about by the alloy composition alone or combined with the influence of strain, when an alloy layer is grown on a virtual substrate of different compositions. In search for direct gap materials, the electronic structure of relaxed or strained Ge1−xSnx and Si1−xSnx alloys, and of strained Ge grown on relaxed Ge1−x−ySixSny, was calculated by the self-consistent pseudo-potential plane wave method, within the mixed-atom supercell model of alloys, which was found to offer a much better accuracy than the virtual crystal approximation. Expressions are given for the direct and indirect band gaps in relaxed Ge1−xSnx, strained Ge grown on relaxed SixGe1−x−ySny and strained Ge1−xSnx grown on a relaxed Ge1−ySny substrate, and these constitute the criteria for achieving a (finite) direct band gap semiconductor. Roughly speaking, good-size (up to ~0.5 eV) direct gap materials are achievable by subjecting Ge or Ge1−xSnx alloy layers to an intermediately large tensile strain, but not excessive because this would result in a small or zero direct gap (detailed criteria are given in the text). Unstrained Ge1−xSnx bulk becomes a direct gap material for Sn content of >17%, but offers only smaller values of the direct gap, typically ≤0.2 eV. On the other hand, relaxed SnxSi1−x alloys do not show a finite direct band gap.