The Gelmke is a small stream, roughly long, and right-hand tributary of the Abzucht in Lower Saxony, Germany. The stream flows through part of the town of Goslar.
The Gelmke rises in the Gelmke Valley (Gelmketal) at about . Its source lies east-northeast of the hill known as the Dicker Kopf (668 m) and north of the Eichenberg (670 m). It flows initially northwards through a densely wooded valley, where it is crossed by the E 11 European long-distance trail before emerging into open fields and swinging east, just above the old Goslar sewage ponds. It then heads north again to join the Abzucht near Goslar station.