Approximable triangulated categories

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