The questions considered belong to the general theme of 'unlikely intersections' but in this talk we focus on various instances in the mod p setting, where p is a large prime.

We will discuss in particular mod p versions of Lang's conjecture on torsion points on varieties and related questions. Generally speaking, in absence of a direct finite field approach, our basic method consists in lifting the problem to a number field and then rely on finiteness results in this setting. The conclusions for individual p are rather weak but much stronger for 'almost all p'.