We report on the deep optical follow-up surveys of the first two gravitational-wave events, GW150914 and GW151226, accomplished by the GRAvitational Wave Inaf TeAm Collaboration (GRAWITA) using the VLT Survey Telescope (VST). We responded promptly to the gravitational-wave alerts sent by the LIGO and Virgo Collaborations, covering a region of $90$ deg$^2$ and $72$ deg$^2$ for GW150914 and GW151226, respectively, and kept observing the two areas over nearly two months. Both surveys reached an average limiting magnitude of about 21 in the $r-$band. The paper outlines the VST observational strategy and two independent procedures developed to search for transient counterpart candidates in multi-epoch VST images. Numerous transients have been discovered, mostly variable stars and eclipsing binaries, but no candidates are identified as related to the gravitational-wave events. The work done let on to gain experience and tune the tools for next LVC runs and in general to exploit the synergies between wide field optical surveys and future multi-messenger programs including projects like Theseus.
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