Pearson's correlation is a measure of the linear relationship between two continuous random variables. It does not assume normality although it does assume finite variances and finite covariance.

Does Pearson correlation assume linearity?

Note: Pearson's correlation determines the degree to which a relationship is linear. Put another way, it determines whether there is a linear component of association between two continuous variables. As such, linearity is not actually an assumption of Pearson's correlation.

What correlation can you use if your data do not meet the assumptions of normal distribution?

In case of non-normal distribution Spearman's correlation coefficient should be used.