  In this survey we present the relatively new concept of \emph{approximable
triangulated categories.} We will show that the definition is natural, that it
leads to powerful new results, and that it throws new light on old, familiar
objects.
  In particular: a recent theorem says that the category $D_{\text{qc}}(X)$ is
approximable whenever $X$ is a quasicompact separated scheme. As corollaries of
this (seemingly technical) statement one can prove striking improvements on old
theorems by Bondal, Rickard, Rouquier and Van den Bergh, about the (much
smaller) categories $D^{\text{perf}}(X)$ and $D^b_{\text{coh}}(X)$.
