  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.
