This is primarily an overview article on some results and problems involving the classical Hardy function $$ Z(t) := ��(1/2+it){\bigl(��(1/2+it)\bigr)}^{-1/2}, \quad ��(s) = ��(s)��(1-s). $$ In particular, we discuss the first and third moment 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.

15 pages. arXiv admin note: text overlap with arXiv:1511.07140