  This paper is a survey article on developments arising from a theorem proved
by Katznelson and Tzafriri in 1986 showing that $\lim_{n\to\infty} \|T^n(I-T)\|
=0$ if $T$ is a power-bounded operator on a Banach space and $\sigma(T) \cap \T
\subseteq \{1\}$. Many variations and consequences of the original theorem have
been proved subsequently, and we provide an account of this branch of operator
theory.
