Soft and compliant robots provide new opportunities for machines that are flexible, adaptable, safe, and robust. Origami-inspired engineering enables custom robots to be designed and fabricated within days, or even hours. These robots are capable of executing a variety of shape-changing and dynamical tasks by taking advantage of their folded shape and programmable mechanics. In this talk, I will show how the kinematics and compliance of a thin-shell mechanism can be designed algorithmically. I will also discuss how we can leverage these designs for better performance and simpler control, and how these approaches provide robots with capabilities such as dynamical hopping, swimming, and flight.