We describe some techniques that can be used to represent and detect deformable shapes in images. The main difficulty with deformable template models is the very large or infinite number of possible nonrigid transformations of the templates. This makes the problem of finding an optimal match of a deformable template to an image incredibly hard. Using a new representation for deformable shapes, we show how to efficiently find a global optimal solution to the nonrigid matching problem. The representation is based on the description of objects using triangulated polygons. Our matching algorithm can minimize a large class of energy functions, making it applicable to a wide range of problems. We present experimental results of detecting shapes in medical images and images of natural scenes. Our method does not depend on initialization and is very robust, yielding good matches even in images with high clutter. We also consider the problem of learning a nonrigid shape model for a class of objects from examples. We show how to learn good models while constraining them to be in the form required by the matching algorithm.