Let $G$ be a finite, simple connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $ρ(G)$ of $G$ is the maximum of the average distances of the vertices of $G$. In this paper, we give sharp upper bounds on the remoteness of a graph of given order, connectivity and size. We also obtain corresponding bound s for $2$-edge-connected and $3$-edge-connected graphs, and bounds in terms of order and size for triangle-free graphs.