Square functions are a fundamental object in harmonic analysis (and also in martingale theory). Since the emergence of the \(H^\infty \) -functional calculus for sectorial operators in the 1980s, deep connections have been forged between this topic and certain square functions, and these links have continued to develop. In this chapter, we introduce square functions tailored to Ritt operators. We show that, under specific geometric conditions on the ambient Banach space, the boundedness of suitable square functions associated with T and \(T^*\) imply that T admits a bounded \(H^\infty \) -functional calculus, and vice versa. We pay particular attention to operators acting either on a Hilbert space, on a Banach lattice with finite cotype, or on a non-commutative \(L^p\) -space.