We introduce CurveFusion, the first approach for high quality scanning of thin structures at interactive rates using a handheld RGBD camera. Thin filament-like structures are mathematically just 1D curves embedded in R³, and integration-based reconstruction works best when depth sequences (from the thin structure parts) are fused using the object's (unknown) curve skeleton. Thus, using the complementary but noisy color and depth channels, CurveFusion first automatically identifies point samples on potential thin structures and groups them into bundles, each being a group of a fixed number of aligned consecutive frames. Then, the algorithm extracts per-bundle skeleton curves using L₁ axes, and aligns and iteratively merges the L₁segments from all the bundles to form the final complete curve skeleton. Thus, unlike previous methods, reconstruction happens via integration along a data-dependent fusion primitive, i.e., the extracted curve skeleton. We extensively evaluate CurveFusion on a range of challenging examples, different scanner and calibration settings, and present high fidelity thin structure reconstructions previously just not possible from raw RGBD sequences.