Abstract: The classical Manin-Mumford conjecture (proved by Raynaud) asserts that if $G$ is a semi-abelian variety over ${\mathbb C}$, $T \leq G({\mathbb C})$ is the torsion subgroup, and $X \subseteq G$ is an irreducible variety with $X({\mathbb C}) \cap T$ Zariski-dense in $X$, then $X$ is a translate by a torsion point of a semi-abelian subvariety of $G$. The direct translation of this statement to positive characteristic is false, but an analogue taking into account varieties defined over finite fields is true. We show: Theorem: Let $K$ be an algebraically closed field of characteristic $p$. Let $G$ be a semi-abelian variety defined over $K$. Let $T \leq G(K)$ be the torsion subgroup. Let $X \subseteq G$ be an irreducible subvariety. If $X(K) \cap T$ is Zariski-dense in $X$, then there is a bijective morphism from $X/{\rm Stab}_G(X)$ to a variety defined over a finite field. In positive characteristic the additive group has non-trivial torsion. We also show an analogue of the Manin-Mumford conjecture for Drinfeld modules (proposed by L. Denis). Theorem: Let $K$ be an algebraically closed field. Let $\varphi: {\mathbb F}_p [t] \to {\rm End}_K {\mathbb G}_a$ be a Drinfeld module of generic characteristic. Let $N \in {\mathbb Z}_+$ be a positive integer and consider $K^N$ as an ${\mathbb F}_p [t]$-module via the diagonal action provided by $\varphi$. Let $T \leq K^N$ be the ${\mathbb F}_p [t]$-torsion submodule. Let $X \subseteq {\mathbb A}_K^N$ be an irreducible subvariety. If $X(K) \cap T$ is Zariski-dense, then $X$ is a translate by a torsion point of an ${\mathbb F}_p [t]$-module. The proofs of these theorems are similar, the main ingredient being a dichotomy theorem in the model theory of difference fields proved by Chatzidakis, Hrushovski, and Peterzil. This method of proof yields uniformity statements for the number of torsion points on varieties and for local heights.