After an introduction which addresses some basic questions, this article is organized around three points: The theoretical framework of the so-called « instrumental approach » which has been a theme in the last

Three conjectural directions in Gromov-Witten theory are discussed: Gorenstein properties, BPS states, and Virasoro constraints. Each points to basic structures in the subject which are not yet understood. 2000 Mathematics Subject Classification: 14N35, 14H10 .

A list of different types of a projective line over non-commutative rings with unity of order up to thirty-one inclusive is given. Eight different types of such a line are found. With a single exception, the basic characteristics of the lines are identical to those of their commutative counterparts. The exceptional projective line is that defined over the non-commutative ring of order sixteen t...