Lectures on the Kato square root problem

This is the text of a series of three lectures given at the CMA of the Australian National University 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 situation 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. I want to thank the CMA at the Australian National University for inviting me during the special program on scattering theory and spectral problems and for the nice and stimulating atmosphere created by the mathematicians at the CMA.