A mapping and/or ablation catheter, comprising: - a tubular member comprising a proximal end, a distal end sized for introduction into a patient's body, a central axis extending therebetween, and a distal portion extending distally from an intermediate portion to the distal end, wherein an electrode is arranged on the distal portion; - an accessory lumen and a steering element lumen extending between the proximal end and the distal end, the steering element lumen offset from the central axis within the distal portion, thereby defining a steering plane intersecting the steering element lumen and the central axis, wherein the accessory lumen is disposed opposite the central axis from the steering element lumen within the distal portion, wherein a conductor extends through the accessory lumen from the proximal end to the distal portion and is coupled to the electrode; - the distal portion comprising a core member extending between the intermediate portion and the distal end comprising a portion of the steering element lumen therein offset from and extending substantially parallel to the central axis, wherein the steering element lumen is arranged above a midline of the core member, and a pair of stabilization elements (50, 50', 150, 350, 450) embedded in the core member creating a stabilization plane (37, 137, 237, 337, 437) extending between the stabilization elements (50, 50', 150, 350, 450) that intersects the steering plane (35, 135, 235, 335, 435), wherein the stabilization elements (50, 50', 150, 350, 450) are arranged below the midline such that the stabilization plane (37, 137, 237, 337, 437) intersects the steering plane (35, 135, 235, 335, 435) at a location closer to the accessory lumen than the steering element lumen; and - a steering element slidably disposed within the steering element lumen comprising a distal end fixed to the tubular member distal end and a proximal end coupled to an actuator on the tubular member proximal end such that, actuation of the actuator applies axial tension or compression to the steering element, thereby causing the distal portion to deflect substantially within the steering plane (35, 135, 235, 335, 435). - a tubular member comprising a proximal end, a distal end sized for introduction into a patient's body, a central axis extending therebetween, and a distal portion extending distally from an intermediate portion to the distal end, wherein an electrode is arranged on the distal portion; - an accessory lumen and a steering element lumen extending between the proximal end and the distal end, the steering element lumen offset from the central axis within the distal portion, thereby defining a steering plane intersecting the steering element lumen and the central axis, wherein the accessory lumen is disposed opposite the central axis from the steering element lumen within the distal portion, wherein a conductor extends through the accessory lumen from the proximal end to the distal portion and is coupled to the electrode; - the distal portion comprising a core member extending between the intermediate portion and the distal end comprising a portion of the steering element lumen therein offset from and extending substantially parallel to the central axis, wherein the steering element lumen is arranged above a midline of the core member, and a pair of stabilization elements (50, 50', 150, 350, 450) embedded in the core member creating a stabilization plane (37, 137, 237, 337, 437) extending between the stabilization elements (50, 50', 150, 350, 450) that intersects the steering plane (35, 135, 235, 335, 435), wherein the stabilization elements (50, 50', 150, 350, 450) are arranged below the midline such that the stabilization plane (37, 137, 237, 337, 437) intersects the steering plane (35, 135, 235, 335, 435) at a location closer to the accessory lumen than the steering element lumen; and - a steering element slidably disposed within the steering element lumen comprising a distal end fixed to the tubular member distal end and a proximal end coupled to an actuator on the tubular member proximal end such that, actuation of the actuator applies axial tension or compression to the steering element, thereby causing the distal portion to deflect substantially within the steering plane (35, 135, 235, 335, 435).