Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If $\overlineσ(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the remoteness of $G$ is defined as the largest value of $\overlineσ(v)$ over all vertices $v$ of $G$. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.