I'll give a proof of Torelli's theorem that curves are determined by their Jacobians. More specifically, suppose $C_1$ and $C_2$ are two (geometrically irreducible smooth projective) curves of genus $g>1$ over a field $k$, and $F:J(C_1)\to J(C_2)$ is any isomorphism of principally polarized abelian varieties. Then there exists an integer $e\in\{\pm 1\}$ and an isomorphism $f:C_1\to C_2$ such that $F=e\cdot J(f)$. If $C_1$, $C_2$ are not hyperelliptic, then the pair $(f,e)$ is determined uniquely by $F$. If they are hyperelliptic, then there are exactly two pairs: $(e,f)$ and $(-e,f\circ\sigma_{C_1}=\sigma_{C_2}\circ f)$, where $\sigma$ denotes the hyperelliptic involution. There is another version of the Torelli theorem which says that $C$ is determined by the function field of $\Sym^{g-1}C$. I may or may not have time to touch on this at the end.