I will discuss recent progress, and open puzzles, in analytical and numerical studies of stochastic fluctuations of line defects on crystal surfaces. From a mathematics viewpoint, this setting helps illuminate the connection of two scales: the nanoscale (where atomic defects are evident) and the macroscale. I will focus on the following issues: (i) The effect of nonlinearities in defect motion; and (ii) the relation between the structure of noise and nanoscale correlations. The mathematical methods involve perturbation theory in the context of a "mean field'' approximation, which I will motivate. The analytical predictions will be compared to kinetic Monte Carlo simulations.