Grasping and manipulating objects is an important human skill. Since contact between hand and object is fundamental to grasping, capturing it can lead to important insights. However, observing contact through external sensors is challenging because of occlusion and the complexity of the human hand. We present ContactDB, a novel dataset of contact maps for household objects that captures the rich hand-object contact during grasping, enabled by use of a thermal camera. Participants in our study grasp 3D printed objects with a post-grasp functional intent. ContactDB includes 3750 3D meshes of 50 household objects textured with contact maps and 375K frames of synchronized RGB-D+thermal images. To the best of our knowledge, this is the first large-scale dataset that records detailed contact maps for functional human grasps. Analysis of this data shows the influence of functional intent and object size on grasping, the tendency to touch/avoid 'active areas' on the object surface, and the importance of palm and lower finger contact. Finally, we learn to predict diverse contact patterns for unseen objects by using state-of-the-art image translation and 3D convolution algorithms.