  We obtain several results and examples concerning the general question ``When
must a space with a small diagonal have a G_delta-diagonal?". In particular, we
show (1) every compact metrizably fibered space with a small diagonal is
metrizable; (2) there are consistent examples of regular Lindelof (even
hereditarily Lindelof) spaces with a small diagonal but no G_delta-diagonal;
(3) every first-countable hereditarily Lindelof space with a small diagonal has
a G_delta-diagonal; (4) assuming CH, every Lindelof Sigma-space with a small
diagonal has a countable network; (5) whether countably compact spaces with a
small diagonal are metrizable depends on your set theory; (6) there is a
locally compact space with a small diagonal but no G_delta diagonal.
