  An $n$ arc in $(k-1)$-dimensional projective space is a set of $n$ points so
that no $k$ lie on a hyperplane. In 1988, Glynn gave a formula to count
$n$-arcs in the projective plane in terms of simpler combinatorial objects
called superfigurations. Several authors have used this formula to count
$n$-arcs in the projective plane for $n \le 10$. In this paper, we determine a
formula to count $n$-arcs in projective 3-space. We then use this formula to
give exact expressions for the number of $n$-arcs in
$\mathbb{P}^3(\mathbb{F}_q)$ for $n \le 7$, which are polynomial in $q$ for $n
\le 6$ and quasipolynomial in $q$ for $n=7$. Lastly, we generalize to
higher-dimensional projective space.
