In one-dimension it is well known that even weak randomness can suppress diffusion and lead to localization. A more quantitative, but still vague, formulation is that systems with local interactions scale W exhibit exponential localization on the scale W^2. For one-dimensional random band matrices, Hermitian matrices with random entries supported in a band of width W around the main diagonal, we show that the eigenvectors decay exponentially at this scale, confirming this prediction. We also discuss progress in this direction for classical analogues.