A ring is called integrally closed if it is an integral domain and is equal to its integral closure in its field of fractions. A scheme is called normal if every stalk is integrally closed.

Some theorems on normality:

A local ring of dimension 1 is normal if and only if it is regular.

(Serre's criterion) A scheme is normal if and only if it is nonsingular in codimension 0 and codimension 1 and every stalk at a generic point of an irreducible closed subset with dimension $\ge 2$ has depth at least 2.

Every rational function on a normal scheme with no poles of codimension 1 is regular.

(Zariski connectedness): If $f:X\rightarrow Y$ is a proper birational map of noetherian integral schemes and $Y$ is normal, then every fiber is connected.

Normal schemes over $C$ are topologically unibranched.

But the proofs I've seen are fairly ad-hoc, and I was wondering if there's some geometric perspective that would clarify these results. The only result here thats an "iff" is Serre's criterion, but I don't understand depth geometrically so I'm not sure how to interpret it.

Is there some nice geometric perspective on normality?