This mini-course will focus on recent advances on the statistical properties of certain number-theoretical sequences, seen as realizations of suitable stochastic processes. We will mainly focus on the distribution of square-free integers and their generalisations (e.g. k-free, B-free). We will discuss Sarnak’s conjecture on the disjointness of the Mobius function \mu(n) from sequences generated by zero-entropy dynamical systems. We will also prove that \mu^2(n) (the indicator of square-free integers) is completely deterministic and study the statistics of its patterns in long intervals.