  Let G be a group and {\phi} be an automorphism of G. Two elements x, y of G
are said to be {\phi}-twisted if y = gx{\phi}(g)^{-1} for some g in G. We say
that a group G has the R_{\infty}-property if the number of {\phi}-twisted
conjugacy classes is infinite for every automorphism {\phi} of G. In this
paper, we prove that twisted Chevalley groups over the field k of
characteristic zero have the R_{\infty}-property as well as S_{\infty}-property
if k has finite transcendence degree over \mathbb{Q} or Aut(k) is periodic.
