I will talk about mod p versions of the Mumford—Tate and André—Oort conjectures. Via a notion of formal linearity, the two conjectures, together with a third one (modpAx—Lindemann), are closely entangled with each other — much closer than their char 0 counterparts. This, for example, reduces the Mumford—Tate conjecture in char p to an “unlikely intersection problem”.  I will talk about my work on establishing the three conjectures for products of orthogonal Shimura varieties. If time permits, I will also go through more recent results beyond the orthogonal case.