Enumerative Invariants in Algebraic Geometry and String Theory : Lectures given at the C.I.M.E. Summer School held in Cetraro Italy June 6–11 2005 Starting in the middle of the 80s there has been a growing and fruitful interaction between algebraic geometry and certain areas of theoretical high energy physics especially the various versions of string theory. Physical heuristics have provided inspiration for new mathematical definitions (such as that of Gromov Witten invariants) leading in turn to the solution of problems in enumerative geometry. Conversely the availability of mathematically rigorous definitions and theorems has benefited the physics research by providing the required evidence in fields where experimental testing seems problematic. The aim of this volume a result of the CIME Summer School held in Cetraro Italy in 2005 is to cover part of the most recent and interesting findings in this subject. Lectures on Gromov–Witten Invariants of Orbifolds Lectures on the Topological Vertex Floer Cohomology with Gerbes The Moduli Space of Curves and Gromov–Witten Theory.