  Let $D$ be a digraph. Its acyclic number $\vec{\alpha}(D)$ is the maximum
order of an acyclic induced subdigraph and its dichromatic number
$\vec{\chi}(D)$ is the least integer $k$ such that $V(D)$ can be partitioned
into $k$ subsets inducing acyclic subdigraphs. We study ${\vec a}(n)$ and $\vec
t(n)$ which are the minimum of $\vec\alpha(D)$ and the maximum of
$\vec{\chi}(D)$, respectively, over all oriented triangle-free graphs of order
$n$. For every $\epsilon>0$ and $n$ large enough, we show $(1/\sqrt{2} -
\epsilon) \sqrt{n\log n} \leq \vec{a}(n) \leq \frac{107}{8} \sqrt n \log n$ and
$\frac{8}{107} \sqrt n/\log n \leq \vec{t}(n) \leq (\sqrt 2 + \epsilon)
\sqrt{n/\log n}$. We also construct an oriented triangle-free graph on 25
vertices with dichromatic number~3, and show that every oriented triangle-free
graph of order at most 17 has dichromatic number at most 2.
