The atom-bond connectivity index ($ABC$) is a vertex-degree based graph invariant, put forward in the 1990s, having applications in chemistry. Let $G=(V,E)$ be a graph, $d_i$ the degree of its vertex $i$, and $ij$ the edge connecting the vertices $i$ and $j$. Then $ABC=\sum_{ij \in E}\sqrt{(d_i+d_j-2)/(d_id_j)}$. Upper bounds and Nordhaus-Gaddum type results for $ABC$ are established.