We propose a new supervized learning framework for oversegmenting 3D point clouds into superpoints. We cast this problem as learning deep embeddings of the local geometry and radiometry of 3D points, such that the border of objects presents high contrasts. The embeddings are computed using a lightweight neural network operating on the points' local neighborhood. Finally, we formulate point cloud oversegmentation as a graph partition problem with respect to the learned embeddings. This new approach allows us to set a new state-of-the-art in point cloud oversegmentation by a significant margin, on a dense indoor dataset (S3DIS) and a sparse outdoor one (vKITTI). Our best solution requires over five times fewer superpoints to reach similar performance than previously published methods on S3DIS. Furthermore, we show that our framework can be used to improve superpoint-based semantic segmentation algorithms, setting a new state-of-the-art for this task as well.