The moduli space $\mathcal{M}_{g,k}(M, J, A)$ consists of equivalence classes of tuples $(\Sigma, j, u, z_1, \ldots, z_k)$ where $(\Sigma, j)$ is a genus-$g$ Riemann surface, $u: \Sigma \to M$ is $J$-holomorphic with $[u] = A \in H_2(M)$, and $z_1, \ldots, z_k \in \Sigma$ are marked points, modulo biholomorphisms of $\Sigma$.

The expected (virtual) dimension of $\mathcal{M}_{g,k}(M, J, A)$ is given by the index formula: $\dim_{\mathrm{vir}} = (n - 3)(2 - 2g) + 2\langle c_1(TM), A \rangle + 2k,$ where $2n = \dim M$. For genus 0 with no marked points: $\dim_{\mathrm{vir}} = 2(n-3) + 2c_1(A)$.

The Gromov-Witten invariant $\mathrm{GW}_{g,k,A}(\alpha_1, \ldots, \alpha_k)$ for cohomology classes $\alpha_i \in H^*(M)$ is defined by: $\mathrm{GW}_{g,k,A}(\alpha_1, \ldots, \alpha_k) = \int_{[\overline{\mathcal{M}}_{g,k}(M,A)]^{\mathrm{vir}}} \mathrm{ev}_1^*\alpha_1 \cup \cdots \cup \mathrm{ev}_k^*\alpha_k,$ where $\mathrm{ev}_i: \overline{\mathcal{M}} \to M$ is the evaluation at the $i$-th marked point and $[\cdots]^{\mathrm{vir}}$ is the virtual fundamental class.