This is the text of a series of three lectures on the recent solution of the square root problem for divergence form elliptic operators, a long-standing conjecture posed by Kato in the early 60’s. In this text, the motivations for this problem and its setting are given. The ideas from harmonic analysis on the T(1) theorem and T(b) theorem for square functions are described. In particular, an apparently new formulation of a local T(b) theorem for square functions is stated. The ideas of the full proof are presented.