This paper develops a Pontryagin Differentiable Programming (PDP) methodology, which establishes a unified framework to solve a broad class of learning and control tasks. The PDP  distinguishes from existing methods by two novel techniques: first, we  differentiate through  Pontryagin's Maximum Principle, and  this allows  to obtain the analytical derivative of a  trajectory with respect to tunable parameters within an optimal control system,  enabling end-to-end learning of   dynamics, policies, or/and control objective functions; and second, we propose an auxiliary control system in the backward pass of the PDP framework, and  the output of this auxiliary control system is the  analytical derivative of the original system's trajectory with respect to the  parameters, which can be iteratively solved using standard control tools. We investigate three learning modes of the PDP: inverse reinforcement learning,  system identification, and  control/planning. We demonstrate the capability of the PDP in each learning mode on different high-dimensional systems, including multi-link robot arm,  6-DoF maneuvering quadrotor, and 6-DoF rocket powered landing.