Over the last twenty years, many people have contributed to a large-scale
programme to study invariants of size across mathematics. At its heart is
the definition of magnitude, a numerical invariant of enriched categories
(including metric spaces and graphs). It unifies size-like invariants such
as groupoid cardinality, Euclidean measure, Euler characteristic, metric
dimension, and entropy. It also comes with its own homology theory that
categorifies it (à la Khovanov), and even a spectral sequence.

The magnitude bibliography currently lists 141 works by 130 authors from
vastly different parts of mathematics and beyond. That's way too much to
cover in one talk, so I'll concentrate on two things: what I view as the
outstanding achievements and the outstanding open questions.