A Hermitian symmetric space is a connected complex manifold with a hermitian metric on which the group of holomorphic isometries acts transitively, and which satisfies the following extra condition: each point (equivalently, some point) is an isolated fixed point of some (holomorphic, isometric) involution of the space. There is a beautiful classification theory of Hermitian symmetric spaces, and this is the starting point for the study of Shimura varieties.

The first four conditions (connected, complex, hermitian, homogeneous), which define a Hermitian homogeneous space, all seem natural enough to me. My question: why the need for the extra condition regarding the involution?

This is deliberately vague because I'm not quite sure what form the most satisfying answer will take. But here are a couple of more specific versions of the question. (1) What sort of spaces are we excluding by imposing this condition, and why? (2) Are there other conditions which might seem more natural to me which are equivalent (or equivalent some of the time)? For instance if you were to tell me that a Hermitian homogeneous space that's negatively curved and simply-connected always has such an involution, I would be completely satisfied.