Some extensions of the Katznelson-Tzafriri theorem

In 1986, Katznelson and Tzafriri proved that, if $T$ is a power-bounded operator on a Banach space $X$, and the spectrum of $T$ meets the unit circle only at 1, then $\|T^n(I-T)\| \to 0$ as $n\to\infty$. Actually, they went further and proved that $\|T^nf(T)\| \to 0$ if $T$ and $f$ satisfy certain conditions. Soon afterward, analogous results were obtained for bounded $C_0$-semigroups $(T(t))_{t\ge0}$. Further extensions and variants were proved later. I will speak about several extensions to the Katznelson-Tzafriri theorem(s), including in particular a recent result(s) obtained by David Seifert and myself.