Modern cars are incorporating an increasing number of driver assist features, among which automatic lane keeping. The latter allows the car to properly position itself within the road lanes, which is also crucial for any subsequent lane departure or trajectory planning decision in fully autonomous cars. Traditional lane detection methods rely on a combination of highly-specialized, hand-crafted features and heuristics, usually followed by post-processing techniques, that are computationally expensive and prone to scalability due to road scene variations. More recent approaches leverage deep learning models, trained for pixel-wise lane segmentation, even when no markings are present in the image due to their big receptive field. Despite their advantages, these methods are limited to detecting a pre-defined, fixed number of lanes, e.g. ego-lanes, and can not cope with lane changes. In this paper, we go beyond the aforementioned limitations and propose to cast the lane detection problem as an instance segmentation problem -in which each lane forms its own instance -that can be trained end-to-end. To parametrize the segmented lane instances before fitting the lane, we further propose to apply a learned perspective transformation, conditioned on the image, in contrast to a fixed "bird's-eye view" transformation. By doing so, we ensure a lane fitting which is robust against road plane changes, unlike existing approaches that rely on a fixed, predefined transformation. In summary, we propose a fast lane detection algorithm, running at 50 fps, which can handle a variable number of lanes and cope with lane changes. We verify our method on the tuSimple dataset and achieve competitive results.