The compensation of motion of the beating heart is investigated in the context of minimally invasive robotic surgery. Although reduced by mechanical stabilisers, residual tissue motion makes surgery still difficult and time consuming. Compensation for this motion is therefore highly desirable. Motion can be captured by tracking natural landmarks on the heart surface viewed by a video endoscope. The proposed motion compensation scenario uses tracked motion information on the beating heart to stabilise image motion at the surgeon's display and to move his instruments accordingly, i.e. synchronously to the surface of the beating heart. The setup uses a specially designed heart motion simulator to display real heart motion and employs a particular prediction strategy to overcome latencies, which is required to achieve synchronous motion of robot instruments and the heart surface. Results show that the proposed methods are suitable to achieve the desired compensation of heart motion.