  We analyze, both theoretically and numerically, the temperature dependent
thermal conductivity \k{appa} of two-dimensional nanowires with surface
roughness. Although each sample is characterized by three independent
parameters - the diameter (width) of the wire, the correlation length and
strength of the surface corrugation - our theory predicts that there exists a
universal regime where \k{appa} is a function of a single combination of all
three model parameters. Numerical simulations of propagation of acoustic
phonons across thin wires confirm this universality and predict a d 1/2
dependence of \k{appa} on the diameter d.
