Enumerative geometry of $K3$ surfaces

The aim of these notes is to explain various enumerative results about $K3$ surfaces without assuming familiarity with Gromov–Witten theory. The enumerative results in question are due to Beauville, Bryan and Leung, Pandharipande, Maulik, Thomas, and others, and confirm conjectures made by Yau–Zaslow, Göttsche, and Katz–Klemm–Vafa.