This talk will focus on recent work motivated by the study of $\lim^n \mathbf{A}$ for $n > 1$. We present results, due to Bergfalk in the case $n = 2$ and Veli\v{c}kovi\'{c}--Vignati in the general case, indicating that each $\lim^n \mathbf{A}$ can consistently be nonzero. We then present complementary results of Bergfalk--Lambie-Hanson and Bergfalk--Hru\v{s}\'{a}k--Lambie-Hanson establishing the consistency of the simultaneous vanishing of the limits $\lim^n \mathbf{A}$ for all $n > 0$. We also discuss recent work of Bannister--Bergfalk--Moore and Bannister extending the techniques of these results to establish the consistency of the additivity of strong homology on a large class of spaces. Time permitting, we will also discuss recent work of Bannister--Bergfalk--Moore--Todorcevic on a new partition relation isolated from the recent work on the vanishing of $\lim^n \mathbf{A}$.