Explore the fascinating world of enumerative geometry in this 58-minute Kirk Public Lecture by Professor Barbara Fantechi from SISSA. Delve into the classical branch of complex projective algebraic geometry that counts solutions to geometric problems, such as the 27 lines on a cubic surface. Discover how ideas from string theory, particularly Gromov-Witten invariants, revolutionized the field in the 1990s, leading to new research directions and ongoing interactions with symplectic geometry, integrable systems, and particle physics. Learn how abstract concepts like schemes naturally arise in concrete problems, and gain insights into the historical development and modern applications of this intriguing mathematical discipline.