The Mobius Function mu(n) is minus one to the number of prime factors of n, if n has no square factors and is zero otherwise. Understanding the randomness (often referred as the Mobius randomness principle) in this function is a fundamental and very difficult problem.We will explain a precise dynamical formulation of this randomness principle and report on recent advances in establishing it and its applications. Tea at 3:30 pm in SC 1425.