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Let $C_1$, $C_2$ be smooth projective curves over an algebraically closed field $K$ of characteristic zero. What is the behavior of the set of non-constant maps $C_1 \to C_2$? Is it infinite, finite, or empty? It turns out that the answer to this question is determined by an invariant of curves called the genus. In particular, if $C_2$ has genus $g(C_2)\geq 2$ (i.e., $C_2$ is hyperbolic), then there are only finitely many non-constant morphisms $C_1 \to C_2$ where $C_1$ is any curve, and moreover, the degree of any map $C_1 \to C_2$ is bounded linearly in $g(C_1)$ by the Riemann--Hurwitz formula.

In this talk, I will explain the above story and discuss a higher dimensional generalization of this result. To this end, I will describe the conjectures of Demailly and Lang which predict a relationship between the geometry of varieties, topological properties of Hom-schemes, and the behavior of rational points on varieties. To conclude, I will sketch a proof of a variant of these conjectures, which roughly says that if $X/K$ is a hyperbolic variety, then for every smooth projective curve $C/K$ of genus $g(C)\geq 0$, the degree of any map $C\to X$ is bounded uniformly in $g(C)$.