  We study the reduction properties of low genus curves whose Jacobian has
complex multiplication. In the elliptic curve case, we classify the possible
Kodaira types of reduction that can occur. Moreover, we investigate the
possible Namikawa Ueno types that can occur for genus $2$ curves whose Jacobian
has complex multiplication which is defined over the base field. We also
produce bounds on the torsion subgroup of abelian varieties with complex
multiplication defined over local fields.
