Structural models used in calculations of properties of substitutionally random ${\mathit{A}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{B}}_{\mathit{x}}$ alloys are usually constructed by randomly occupying each of the N sites of a periodic cell by A or B. We show that it is possible to design ``special quasirandom structures'' (SQS's) that mimic for small N (even N=8) the first few, physically most relevant radial correlation functions of a perfectly random structure far better than the standard technique does. We demonstrate the usefulness of these SQS's by calculating optical and thermodynamic properties of a number of semiconductor alloys in the local-density formalism.