Spaces having a small diagonal

We obtain several results and examples concerning the general question “When must a space with a small diagonal have a Gδ-diagonal?”. In particular, we show compact metrizably fibered space with a small diagonal is metrizable; are consistent examples of regular Lindelöf (even hereditarily Lindelöf) spaces with a small diagonal but no Gδ-diagonal; first-countable hereditarily Lindelöf space with a small diagonal has a Gδ-diagonal; ng CH, every Lindelöf Σ-space with a small diagonal has a countable network; atement “countably compact spaces with a small diagonal are metrizable” is consistent with and independent of ZFC; is in ZFC a locally compact space with a small diagonal but no Gδ-diagonal.