This text has grown out of a mini-course held at the Arctic Number Theory School, University of Helsinki, May 18–25, 2011. The centralt opic is Hardy's function Z(t), of great importance in the theory of the Riemann zeta-function ζ(s). It is named after Godfrey Harold (“G. H.”) Hardy FRS (1877–1947), who was a prominent English mathematician, well-known for his achievements in number theory and mathematical analysis. Sometimes by Hardy function(s) one denotes the element(s) of Hardys paces Hp, which are certain spaces of holomorphic functions on the unit disk or the upper half-plane. In this text, however, Hardy's function Z(t) will always denote the function defined by (0) below. It was chosen as the object of study because of its significance in the theory of ζ(s) and because, initially, considerable material could be presented on the blackboard within the f ramework of six lectures. Some results, like Theorem 6.7 and the bounds in (4.25) and (4.26) are new, improving on older ones. It is “Hardy's function” which is the thread that holds this work together. I have thought it is appropriate for a monograph because the topic is not as vast as the topic of the Riemann zeta-function itself.