Both $\omega_1$ and $-\omega_1$ have the property that they are minimal uncountable linear orders --- they embed into all of their uncountable suborders. It has long been known that in some models of set theory, there are other minimal uncountable linear orders. For instance Baumgartner demonstrated that it is consistent that any subset of the real line of cardinality $\aleph_1$ is a minimal uncountable linear order. In this talk I will outline the proof of a result of mine from 2007: it is consistent that $\omega_1$ and $-\omega_1$ are the only minimal uncountable linear orders.