After proving the existence of Riemannian metrics on manifolds, one of the students asked if the "paracompactness" is necessary. Of course the standard proof with the partition of unity uses this assumption, and for the only non-paracompact manifold I know, namely the long line, a similar argument seems to show the existence of a metric. So, just out of curiosity, does there exist a "manifold" (of course, non-paracomapact) which does not admit any Riemannian metric?
On the contrary, the long line does not have a Riemannian metric. Every countable subset of the long line has a least upper bound, so if it were Riemannian then a geodesic ray in the long direction would have to stop short of the end. But then the Riemannian metric would break down at the "endpoint" of this geodesic.
More generally, it is a theorem of Stone that every metric space is paracompact, hence every Riemannian metric space is too.