  It is well-known that real data often contain outliers. The term outlier
typically refers to a case, that is, a row of the $n \times d$ data matrix. In
recent times a different type has come into focus, the cellwise outliers. These
are suspicious cells (entries) that can occur anywhere in the data matrix. Even
a relatively small proportion of outlying cells can contaminate over half the
rows, which is a problem for rowwise robust methods. In this article we discuss
the challenges posed by cellwise outliers, and some methods developed so far to
deal with them. We obtain new results on cellwise breakdown values for
location, covariance and regression. We also propose a cellwise robust method
for correspondence analysis, with real data illustrations. The paper concludes
by formulating some points for debate.
