This is an introduction to enumerative geometry, mirror symmetry, and topics beyond. Gromov Witten theory (GW) enumerates the number of Riemann surfaces in Calabi Yau (CY) manifolds. It originated from IIA string theory and became celebrated in enumerative algebraic geometry. Physics predicted GW theory is equivalent to type IIB string theory on another CY manifolds called Mirror. Such prediction generates a lot of research directions. We shall introduce both the GW theory and mirror symmetry conjecture. In the end of the course we shall include subjects such as bosonic strings or Chern Simons theories, with an eye to peek into the non-enumerative realm. To take this course, one is required to know basic concepts in Undergraduate mathematics such as the definitions of Riemann surfaces, sheaf cohomology and Chern classes. The language of schemes will be used throughout the course. Backgrounds in path integrals or QFT will be helpful.