A central research challenge for the
mathematical sciences in the $21^{st}$ century is
the development of principled methodologies for the
seamless integration of (often vast) data sets
with (often sophisticated) mathematical models.
Such data sets are becoming routinely available
in almost all areas of engineering, science and technology,
whilst mathematical models describing phenomena of
interest are often built on decades, or even centuries, of
human knowledge creation. Ignoring either the data or the models
is clearly unwise and so the issue of combining them
is of paramount importance. In this talk we will give
a historical perspective on the subject, highlight some of
the current research directions that it leads to, and
describe some of the underlying mathematical frameworks
begin deployed and developed. The ideas will be illustrated
by problems arising in the geophysical, biomedical and
social sciences.