  In this paper, we show that Hilbert transforms along some curves are bounded
on $L^p({\mathbb R}^n;X)$ for some $1<p<\infty$ and some UMD spaces $X$. In
particular, we prove that the Hilbert transform along some curves are
completely $L^p$-bounded in the terminology from operator space theory.
Moreover, we obtain the $L^p(\mathbb{R}^n;X)$-boundedness of anisotropic
singular integrals by using the "method of rotations" of Calder\'{o}n-Zygmund.
  All these results extend the existing related ones.
