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We construct the Atiyah classes of holomorphic vector bundles using (1, 0)-connections and developing a Chern-Weil type theory, allowing us to effectively compare Chern and Atiyah forms. Combining this point of view with the Čech-Dolbeault cohomology, we prove several results about vanishing and localization of Atiyah classes, and give some applications. In particular, we prove a Bott type vanishing theorem for (not necessarily involutive) holomorphic distributions. As an example we alsodoi:10.4171/rmi/730 fatcat:kc55qwr5lfafjjwgtpnpgmtfxm