  Recently, it was realized that anomalies can be completely classified by
topological orders, symmetry protected topological (SPT) orders, and symmetry
enriched topological orders in one higher dimension. The anomalies that people
used to study are invertible anomalies that correspond to invertible
topological orders and/or symmetry protected topological orders in one higher
dimension. In this paper, we introduce a notion of non-invertible anomaly,
which describes the boundary of generic topological order. A key feature of
non-invertible anomaly is that it has several partition functions. Under the
mapping class group transformation of space-time, those partition functions
transform in a certain way characterized by the data of the corresponding
topological order in one higher dimension. In fact, the anomalous partition
functions transform in the same way as the degenerate ground states of the
corresponding topological order in one higher dimension. This general theory of
non-invertible anomaly may have wide applications. As an example, we show that
the irreducible gapless boundary of 2+1D double-semion (DS) topological order
must have central charge $c=\bar c \geq \frac{25}{28}$.
