A famous theorem by Siu states that plurigenera are deformation invariant for families of smooth complex algebraic varieties. This result, and the techniques involved in its proof, have had a big impact in various developments of higher dimensional algebraic geometry. In this talk we will address a positive characteristic analogue of the aforementioned result: first we will see how to recover an “up to $p$-power” version of Siu’s theorem assuming the MMP and Abundance conjecture, extending a result of Nakayama. Then we will present some examples showing a strong failure of invariance of plurigenera in positive characteristic. Finally, I will explain how to recover Siu’s theorem for deformation of good minimal models whose Iitaka fibration satisfies certain “ordinarity” assumptions