What is the relationship between independence and correlation?

If two random variables X and Y are independent, it is always true that they are uncorrelated, meaning that if X and Y are independent, corr(X,Y) = 0.

However, the converse is not necessarily true, meaning that if corr(X,Y)=0, X and Y are not guaranteed to be independent of one another. This is because correlation is a measure of linear association, and while a correlation of 0 implies there is no linear relationship between X and Y, there might be a more complicated higher order relationship that the correlation coefficient would not accurately quantify.