I will discuss Kodaira vanishing and its generalization, the Kawamata–Viehweg vanishing theorem. Kodaira vanishing fails in general in positive characteristic, already for smooth surfaces. I will present special situations in which these theorems still hold for certain classes of surfaces, and explain how such cases can sometimes be leveraged to recover birational versions of vanishing for threefolds. I will also highlight why these results matter for the minimal model program. The talk will survey key themes in my research to date and aims to spark interest in these topics within the group.