1) *Verlinde's formula*: Let $\mathrm{C}$ be a curve of genus $g\geqslant 2$. The Picard group of the Moduli space of rank $2$ bundles on $\mathrm{C}$ with trivial determinant is an infinite cyclic group, with ample generator denoted by $\mathscr{L}$. Verlinde's formula says that $$h^0(\mathscr{L}^{\otimes k})=\sum_{p=0}^{k} \mathrm{S}_{p0}^{-\chi(\mathrm{C})},$$
where the matrix $\mathrm{S}=(\mathrm{S}_{pq})_{p,q=0}^{k}$ is given by
$$\mathrm{S}_{pq}=\sqrt{\frac{2}{k+2}}\sin\frac{(p+1)(q+1)}{k+2}\pi.$$
This was proven in the early 90s by several people (Bertram-Szenes, Faltings, Thaddeus,...).
The corresponding result for rank $1$ bundles (in which case the right hand side of the formula is $k^g$) is classical and easy to prove. For rank $2$ bundles the Moduli space is not smooth, let alone a torus, and so the intersection theory is very hard to determine.

2) *Witten's conjecture*: Consider the DM compactified moduli space $\overline{\mathscr{M}}_{g,n}$ of genus $g$ curves with $n$ markings. For each $1\leqslant i\leqslant n$ there is a line bundle $\mathscr{L}_i$ on $\overline{\mathscr{M}}_{g,n}$ whose fiber over $(\mathrm{C};p_1,\ldots,p_n)$ is the cotangent space $\mathrm{T}^\vee_{\mathrm{C},p_i}$. Let $\psi_i=c_1(\mathscr{L}_i)$. For $k_1,\ldots,k_n\geqslant 0$ define $\left\langle\tau_{k_1}\cdots\tau_{k_n}\right\rangle_g$ to be
$$\int_{\overline{\mathscr{M}}_{g,n}}\psi_1^{k_1}\cdots\psi_n^{k_n}$$
if $\sum_{i=1}^n k_i=\dim\overline{\mathscr{M}}_{g,n}=3g-3+n$ and $0$ otherwise. Then define
$$F_g((t_i)_{i=0}^\infty)=\sum_d \left(\prod_{i=1}^\infty \frac{t_i^{d_i}}{d_i!}\right)\left\langle\tau_0^{d_0}\tau_1^{d_1}\tau_2^{d_2}\cdots\right\rangle_g,$$
the summation being over all sequences $d=(d_i)_{i=1}^\infty$ of natural numbers with finite support. Consider the generating series
$$F=\sum_{g=0}^\infty F_g \lambda^{2g-2}$$
with derivatives
$$\left\langle\left\langle\tau_{k_1}\cdots\tau_{k_n}\right\rangle\right\rangle=\frac{\partial}{\partial t_{k_1}}\cdots\frac{\partial}{\partial t_{k_n}}F.$$
Witten's conjecture says that for all $n\geqslant 1$
$$(2n+1)\lambda^{-2}\left\langle\left\langle\tau_n \tau_0^2\right\rangle\right\rangle=\left\langle\left\langle\tau_{n-1}\tau_0\right\rangle\right\rangle\left\langle\left\langle\tau_0^3\right\rangle\right\rangle+2\left\langle\left\langle\tau_{n-1} \tau_0^2\right\rangle\right\rangle\left\langle\left\langle\tau_0^2\right\rangle\right\rangle+\frac{1}{4}\left\langle\left\langle\tau_{n-1} \tau_0^4\right\rangle\right\rangle.$$
(For more on this see for example Harris & Morrison, Moduli of Curves, page 71.)
For $n=1$ Witten's conjecture means that $U=\partial^2 F/\partial t_0^2$ satisfies the KdV equation
$$3\lambda^{-2}\frac{\partial U}{\partial t_1}=3U\frac{\partial U}{\partial t_0}+\frac{1}{4}\frac{\partial^3 U}{\partial t_0^3}.$$
Witten's conjecture was first proven by Kontsevich, and again several other people have given alternative proofs (Okounkov-Pandharipande, Kazarian-Lando, Mirzakhani,...). But I find it difficult to imagine that this result would have seen the light of day without string theory.

3) *Counting rational curves on quintic threefolds*, see for example Pandharipande's Séminaire Bourbaki talk. Some of the history is described here.