  We provide a definition of Tanaka-Thomas's Vafa-Witten invariants for \'etale
gerbes over smooth projective surfaces using the moduli spaces of $\mu_r$-gerbe
twisted sheaves and Higgs sheaves. Twisted sheaves and their moduli are
naturally used to study the period-index theorem for the corresponding
$\mu_r$-gerbe in the Brauer group of the surfaces. Deformation and obstruction
theory of the twisted sheaves and Higgs sheaves behave like general sheaves and
Higgs sheaves. We define virtual fundamental classes on the moduli spaces and
define the twisted Vafa-Witten invariants using virtual localization and the
Behrend function on the moduli spaces. As applications for the Langlands dual
group $\SU(r)/\zz_r$ of $\SU(r)$, we define the $\SU(r)/\zz_r$-Vafa-Witten
invariants using the twisted invariants for \'etale gerbes, and prove the
S-duality conjecture of Vafa-Witten for the projective plane in rank two and
for K3 surfaces in prime ranks. We also conjecture for other surfaces.
