Asymptotic free independence ensures that both individual and mixed moments of first order exist and provides us with universal rules to calculate mixed moments of first order out of individual ones. One might then wonder if there are universal rules for computing mixed moments of higher

order out individual ones. The most studied moments of higher order are moments of second order, also known as fluctuation moments.

In this talk, we first recollect some of the most relevant results on fluctuation moments, and then present some examples of random matrices exhibiting distinct and new fluctuation moments behavior.