The covering number, $σ(G)$, of a finite, noncyclic group $G$ is the least positive integer $n$ such that $G$ is the union of $n$ proper subgroups. Here we investigate the covering numbers of the projective special unitary groups $U_3(q)$, give upper and lower bounds for $σ(U_3(q))$ when $q \geq 7$, and show that $σ(U_3(q))$ is asymptotic to $q^6/3$ as $q \rightarrow \infty$.