The tasks of image registration (i.e., the computation of correspondences) and image-based reconstruction (i.e., the computation of depth) are fundamental in Computer Vision. Solving both tasks is required in fields such as video editing, robotics, or Augmented Reality (AR) applications. To date, there exist mature techniques for the reconstruction and registration of rigid objects, such as Structure-from-Motion (SfM). The case of deformable objects is however largely unresolved, despite being crucial in applications where most of the objects are non-rigid, such as in medical imaging. There exist two main scenarios. In Non-Rigid SfM (NRSfM), the inputs are a set of images and the problem is to find correspondences across images (registration) and depth (reconstruction). It is thus an extension of SfM to non-rigid objects. In Shape-from-Template (SfT), the inputs are a single image and a 3D object model (template), and the problem is to find correspondences between the model and the image (registration) and depth (reconstruction). Obviously, as the object is deformable, the image is not a photo of the model under some unknown pose: rather, it is a photo of the model taken after some unknown deformation. In the last few years, both SfT and NRSfM have been thoroughly for some types of deformation models that can be expressed as partial differential equations, such as the isometric model. Based on those models, NRSfM and SfT can be solved analytically. This talk gives insight into these methods from both a theoretical and practical perspective.