The well-established single-molecule force-spectroscopy techniques have recently been complemented by methods that can measure torque and twist directly, notably magnetic torque tweezers and the optical torque wrench. A limitation of the current torque measurement schemes is the intrinsic coupling between the force and torque degrees of freedom. Here we present electromagnetic torque tweezers (eMTT) that combine permanent and electromagnets to enable independent control of the force and torsional trap stiffness for sensitive measurements of single molecule torque and twist. Using the eMTT, we demonstrate sensitive torque measurements on tethered DNA molecules from simple tracking of the beads' (x,y)-position, obviating the need for any angular tracking algorithms or markers. Employing the eMTT for high-resolution torque measurements, we experimentally confirm the theoretically predicted torque overshoot at the DNA buckling transition in high salt conditions. We envision that the flexibility and control afforded by the eMTT will enable a range of new torque and twist measurement schemes from single-molecules to living cells.