  A (unit) disk graph is the intersection graph of closed (unit) disks in the
plane. Almost three decades ago, an elegant polynomial-time algorithm was found
for \textsc{Maximum Clique} on unit disk graphs [Clark, Colbourn, Johnson;
Discrete Mathematics '90]. Since then, it has been an intriguing open question
whether or not tractability can be extended to general disk graphs. We show the
rather surprising structural result that a disjoint union of cycles is the
complement of a disk graph if and only if at most one of those cycles is of odd
length. From that, we derive the first QPTAS and subexponential algorithm
running in time $2^{\tilde{O}(n^{2/3})}$ for \textsc{Maximum Clique} on disk
graphs. In stark contrast, \textsc{Maximum Clique} on intersection graphs of
filled ellipses or filled triangles is unlikely to have such algorithms, even
when the ellipses are close to unit disks. Indeed, we show that there is a
constant approximation which is not attainable even in time
$2^{n^{1-\varepsilon}}$, unless the Exponential Time Hypothesis fails.
