Strong generators in $D^{perf}(X)$ and $D_{coh}^b(X)$

We characterize the separated schemes for which the derived category of perfect complexes is strongly generated, proving a conjecture of Bondal and Van den Bergh. And we also prove the strong generation of the bounded derived category of coherent sheaves under mild conditions. In the proof we introduce techniques that have shown themselves to have powerful consequences in other contexts. In this final, pre-publication version of the manuscript there is a (very brief) survey of subsequent, followup articles.