Geometric Manin's conjecture in characteristic $p$

Geometric Manin’s conjecture for complex Fano varieties describes the structure of the moduli space of curves. We propose a version of this conjecture in characteristic $p$ and describe its connection to the Batyrev–Manin–Peyre–Tschinkel conjecture over global fields. This is a survey paper written for a volume of the Summer Research Institute in Algebraic Geometry held at Colorado State University in 2025.