  We study varieties $X \subseteq \mathbb P^N$ of dimension $n$ such that
$T_X(k)$ is an Ulrich vector bundle for some $k \in \mathbb Z$. First we give a
sharp bound for $k$ in the case of curves. Then we show that $k \le n+1$ if $2
\le n \le 12$. We classify the pairs $(X,\mathcal O_X(1))$ for $k=1$ and we
show that, for $n \ge 4$, the case $k=2$ does not occur.
