# What does vanishing of higher direct images of the structure sheaf tell us?

Let $$f:X\to Y$$ be a morphism of schemes over $$k$$. I am wondering about, what geometric consequences $$R^qf_*O_X=0$$ for $$q\geq k$$ does have. I saw vanishing of higher direct images used in some proofs of coherence of structure sheaves and also for the computation of cohomology if one in addition assumes that $$f_*O_X=O_Y$$. Also there seem to be a lot of papers out there discussing cases in which the higher direct images vanish for $$q\geq 1$$.

What are natural situations in which one wants vanishing of higher direct images and what does it tell us (geometrically)?

I know this is kind of vague but I was unable to figure it out myself. Thanks in advance!