Automotive radar is a promising technology with regard to path planning, since radar systems offer a comparatively long range and are robust against bad weather conditions. In this paper, we use Convolutional Neural Networks (CNN) to determine the current road course from radar point clouds. For this purpose, we first transform the radar point cloud into various gridmaps, which then serve as an input for the CNN. The quality of the road course estimation is evaluated using a test dataset. Exemplary test results showed an average deviation of less than 91 cm at a range of 100 m between the ground truth and the estimated road course. These excellent results prove that CNN processing of radar measurements is an attractive option for reliable and precise road course estimation.