We present Shape-Tailored Deep Neural Networks (ST-DNN). ST-DNN are deep networks formulated through the use of partial differential equations (PDE) to be defined on arbitrarily shaped regions. This is natural for problems in computer vision such as segmentation, where descriptors should describe regions (e.g., of objects) that have diverse shape. We formulate ST-DNNs through the Poisson PDE, which can be used to generalize convolution to arbitrary regions. We stack multiple PDE layers to generalize a deep CNN to arbitrarily shaped regions. We show that ST-DNN are provably covariant to translations and rotations and robust to domain deformations, which are important properties for computer vision tasks. We show proof-of-concept empirical validation.