Lawrence D. Guth continues MIT's 18.156 Projection Theory course with the third installment on random walks on finite groups, part of a series examining how projection and incidence methods connect to group theory and combinatorics. This session focuses on the Bourgain-Gamburd results, a landmark set of theorems on spectral gaps for random walks on finite groups such as SL2 of a finite field, and walks through some of the key proof ideas behind them, including how additive combinatorics and expansion properties interact. Guth develops the material on the board in a standard graduate lecture format, building on the previous two sessions in the sequence. The lecture assumes familiarity with group theory and prior lectures in the course and is aimed at graduate students or researchers working in combinatorics, number theory, or geometric measure theory.