TITLE: Why are non-separated schemes schemes?
QUESTION [6 upvotes]: In "the old days", e.g. in the famous texts by Grothendieck and Mumford, a scheme was defined as what we now call a separated scheme. (i.e. a scheme where the image of the morphism $\Delta:X \to X \times X$ is closed)
Nowadays, schemes are usually allowed to be separated. The question is then: Do non-separated schemes naturally occur in nature?

REPLY [4 votes]: Non-separated schemes also appear in GIT; i.e., when you want to define the quotient of a scheme by a group of automorphisms.