We propose an identity (ID)-based signature from m-torsion groups of super-singular elliptic curves or hyper-elliptic curves. It is secure against existential forgery under adaptively chosen message and ID attack in the random oracle model. Due to the nice properties of m-torsion groups and Weil pairing, it turns out that the proposed scheme is simple, efficient and has the good property of addition. Based on this ID-based signature scheme, we propose an ID-based threshold signature scheme and show that it has the security properties of unforgeability and robustness.