Twistor spaces are 3-dimensional complex manifolds which are naturally associated to anti-self-dual (ASD) conformal structures on 4-manifolds. In this talk, first I briefly recall generalities on ASD structures and the associated twistor spaces, and next survey some classical results on compact Kaehler/Moishezon twistor spaces that were obtained by 1990s. Then I will describe more recent development concerning classification and new examples of Moishezon twistor spaces.
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