We address the curve lane detection problem which poses more realistic challenges than conventional lane detection for better facilitating modern assisted/autonomous driving systems. Current hand-designed lane detection methods are not robust enough to capture the curve lanes especially the remote parts due to the lack of modeling both long-range contextual information and detailed curve trajectory. In this paper, we propose a novel lane-sensitive architecture search framework named CurveLane-NAS to automatically capture both long-ranged coherent and accurate short-range curve information. It consists of three search modules: a) a feature fusion search module to find a better fusion of the local and global context for multi-level hierarchy features; b) an elastic backbone search module to explore an efficient feature extractor with good semantics and latency; c) an adaptive point blending module to search a multi-level post-processing refinement strategy to combine multi-scale head prediction. Furthermore, we also steer forward to release a more challenging benchmark named CurveLanes for addressing the most difficult curve lanes. It consists of 150K images with 680K labels (The new dataset can be downloaded at http://www.noahlab.com.hk/opensource/vega/#curvelanes ). Experiments on the new CurveLanes show that the SOTA lane detection methods suffer substantial performance drop while our model can still reach an 80+% F1-score. Extensive experiments on traditional lane benchmarks such as CULane also demonstrate the superiority of our CurveLane-NAS, e.g. achieving a new SOTA 74.8% F1-score on CULane.