  An arc in $\mathbb F_q^2$ is a set $P \subset \mathbb F_q^2$ such that no
three points of $P$ are collinear. We use the method of hypergraph containers
to prove several counting results for arcs. Let $\mathcal A(q)$ denote the
family of all arcs in $\mathbb F_q^2$. Our main result is the bound \[
|\mathcal A(q)| \leq 2^{(1+o(1))q}. \] This matches, up to the factor hidden in
the $o(1)$ notation, the trivial lower bound that comes from considering all
subsets of an arc of size $q$.
  We also give upper bounds for the number of arcs of a fixed (large) size. Let
$k=q^t$ for some $t >2/3$, and let $\mathcal A(q,k)$ denote the family of all
arcs in $\mathbb F_q^2$ with cardinality $k$. We prove that, for all $\gamma
>0$ \[ |\mathcal A(q,k)| \leq \binom{(1+\gamma)q}{k}. \] This result improves a
bound of Roche-Newton and Warren. A nearly matching lower bound \[ |\mathcal
A(q,k)| \geq \binom{q}{k} \] follows by considering all subsets of size $k$ of
an arc of size $q$.
