Not yet reviewed by Pith; the record is open.

The conditions for the cancellation of all gauge, gravitational, and mixed anomalies of $N=1$ supersymmetric models in six dimensions are reviewed and illustrated by a number of examples. Of particular interest are models that cannot be realized perturbatively in string theory. An example of this type, which we verify satisfies the anomaly cancellation conditions, is the K3 compactification of the $SO(32)$ theory with small instantons recently proposed by Witten. When the instantons coincide it has gauge group $SO(32) \times Sp(24)$. Two new classes of models, for which non-perturbative string constructions are not yet known, are also presented. They have gauge groups $SO(2n+8)\times Sp(n)$ and $SU(n)\times SU(n)$, where $n$ is an arbitrary positive integer.