Active Surface Models have a long history of being useful to model complex 3D surfaces. But only Active Contours have been used in conjunction with deep networks, and then only to produce the data term as well as meta-parameter maps controlling them. In this paper, we advocate a much tighter integration. We introduce layers that implement them that can be integrated seamlessly into Graph Convolutional Networks to enforce sophisticated smoothness priors at an acceptable computational cost. We will show that the resulting Deep Active Surface Models outperform equivalent architectures that use traditional regularization loss terms to impose smoothness priors for 3D surface reconstruction from 2D images and for 3D volume segmentation.