Abstract: The Mobius function is one of the most important arithmetic functions. There is a vague yet well known principle regarding its randomness properties called the “Mobius randomness law". It basically states that the Mobius function should be orthogonal to any "structured" sequence. P. Sarnak suggested a far reaching conjecture as a possible formalization of this principle. He conjectured that "structured sequences" should correspond to sequences arising from deterministic dynamical systems. Sarnak’s conjecture follows from Chowla’s conjecture - which is the mobius version of the prime tuple conjecture. I will describe progress in recent years towards these conjectures, building on major advances dynamics, additive combinatorics, and analytic number theory.