  We report the results of deep optical follow-up surveys of the first two
gravitational-wave sources, GW150914 and GW151226, done by the GRAvitational
Wave Inaf TeAm Collaboration (GRAWITA). The VLT Survey Telescope (VST)
responded promptly to the gravitational-wave alerts sent by the LIGO and Virgo
Collaborations, monitoring a region of $90$ deg$^2$ and $72$ deg$^2$ for
GW150914 and GW151226, respectively, and repeated the observations over nearly
two months. Both surveys reached an average limiting magnitude of about 21 in
the $r-$band. The paper describes the VST observational strategy and two
independent procedures developed to search for transient counterpart candidates
in multi-epoch VST images. Several transients have been discovered but no
candidates are recognized to be related to the gravitational-wave events.
Interestingly, among many contaminant supernovae, we find a possible
correlation between the supernova VSTJ57.77559-59.13990 and GRB150827A detected
by {\it Fermi}-GBM. The detection efficiency of VST observations for different
types of electromagnetic counterparts of gravitational-wave events are
evaluated for the present and future follow-up surveys.
