Random band matrices provide a simple model for studying the metal-insulator transition of random operators, the prominent example of which is the Anderson model. According to the famous sqrt(N) conjecture, N x N random matrices with non-zero entries only within a diagonal band of width W exhibit a phase transition between GUE behavior and Anderson localization, precisely at W ~ sqrt(N). After giving some background on this problem, we will first present a unique special “chiral” model where the conjecture can be shown to hold, and then proceed to study the full model, proving localization holds at all energies when W ≪ N^{1/4}. This latter result uses the fractional moment method and an adaptive Mermin-Wagner-style shift.