In 1972, the late B. E. Johnson introduced the notion of an amenable Banach
algebra and asked whether the Banach algebra $B(E)$ of all bounded linear
operators on a Banach space $E$ could ever be amenable if $\dim E = \infty$.
Somewhat surprisingly, this question was answered positively only very recently
as a by-product of the Argyros--Haydon result that solves the "scalar plus
compact problem": there is an infinite-dimensional Banach space $E$, the dual
of which is $\ell^1$, such that $B(E) = K(E)+ \mathbb{C} \, \id_E$. Still,
$B(\ell^2)$ is not amenable, and in the past decade, $ B(\ell^p)$ was found to
be non-amenable for $p=1,2,\infty$ thanks to the work of C. J. Read, G. Pisier,
and N. Ozawa. We survey those results, and then---based on joint work with M.
Daws---outline a proof that establishes the non-amenability of $B(\ell^p)$ for
all $p \in [1,\infty]$.