The primary theme of this thesis is the study of recursion formulae in enumerative geometry. In Chapter 2 we define moduli spaces of relative stable quasimaps in genus zero, and derive a recursion formula which allows us to compute the resulting relative quasimap invariants. We apply this formula to obtain a quantum Lefschetz theorem for quasimap invariants (this is joint work with Luca Battistella). In Chapter 3 we present work in progress towards a recursion formula for log Gromov–Witten invariants. Along the way, we introduce auxiliary moduli spaces and use them to probe the geometry of the moduli space of log stable maps. Finally in Chapter 4, we express a fundamental object in ordinary Gromov–Witten theory – Givental’s Lagrangian cone – using relative stable maps. As a corollary, we obtain a sequence of universal relations involving the Gromov–Witten invariants.