H∞ functional calculus, introduced by Alan McIntosh in the 80s and developed in collaboration with M. Cowling, I. Doust and A. Yagi, plays an important role in several branches of operator theory. The main topic treated in the book is natural square functions associated with sectorial operators, or with semigroups, acting on suitable non-commutative Lp-spaces and their relations with H∞ functional calculus. After providing a background on non-commutative Lp-spaces, H∞ functional calculus, sectorial operators and semigroups, the authors introduce square functions and describe their relations to H∞ functional calculus. There is a chapter devoted to a noncommutative generalization of the Stein diffusion semigroup. Further topics include multiplication operators, Hamiltonians and the Schur multipliers on Schatten space, semigroups on q-deformed von Neumann algebras, and the noncommutative Poisson semigroup of a free group. The last chapter briefly treats the non tracial case.