  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.
