  Over all graphs (or unicyclic graphs) of a given order, we characterise those
graphs that minimise or maximise the number of connected induced subgraphs. For
each of these classes, we find that the graphs that minimise the number of
connected induced subgraphs coincide with those that are known to maximise the
Wiener index (the sum of the distances between all unordered pairs of
vertices), and vice versa. For every $k$, we also determine the connected
graphs that are extremal with respect to the number of $k$-vertex connected
induced subgraphs. We show that, in contrast to the minimum which is uniquely
realised by the path, the maximum value is attained by a rich class of
connected graphs.
