Hardy's function $Z(t)$: Results and problems

This is primarily an overview article on some results and problems involving the classical Hardy function $Z(t) := \zeta (1/2+it)(\chi (1/2+it))^{-1/2}$, $\zeta (s) = \chi (s)\zeta (1-s)$. In particular, we discuss the first and third moments of $Z(t)$ (with and without shifts) and the distribution of its positive and negative values. A new result involving the distribution of its values is presented.