The difficulty of obtaining annotations to build training databases still slows down the adoption of recent deep learning approaches for biomedical image analysis. In this paper, we show that we can train a Deep Net to perform 3D volumetric delineation given only 2D annotations in Maximum Intensity Projections (MIP) of the training volumes. This significantly reduces the annotation time: We conducted a user study that suggests that annotating 2D projections is on average twice as fast as annotating the original 3D volumes. Our technical contribution is a loss function that evaluates a 3D prediction against annotations of 2D projections. It is inspired by space carving, a classical approach to reconstructing complex 3D shapes from arbitrarily-positioned cameras. It can be used to train any deep network with volumetric output, without the need to change the network's architecture. Substituting the loss is all it takes to enable 2D annotations in an existing training setup. In extensive experiments on 3D light microscopy images of neurons and retinal blood vessels, and on Magnetic Resonance Angiography (MRA) brain scans, we show that, when trained on projection annotations, deep delineation networks perform as well as when they are trained using costlier 3D annotations. (C) 2019 Elsevier B.V. All rights reserved.