How robust is Pearson's Correlation to Normality violations?

Hi Guys,
I can't seem to find a definitive answer to this question: can one run a Pearson's correlation on data that is not normal (but has no outliers). Some texts say you need to obey this assumption, other authors say that it is robust.

The problem is, is that my data is negatively skewed and my significant finding disappears when I use a log transformation or when I use Kendall's tau/ Spearman.

Any a advice? Or perhaps a reference I can look up?
Many thanks!