Model Predictive Control (MPC) promises to endow robots with enough reactivity to perform complex tasks in dynamic environments by frequently updating their motion plan based on measurements. Despite its appeal, it has seldom been deployed on real machines because of scaling constraints. This paper presents the first hardware implementation of closed-loop nonlinear MPC on a 7-DoF torque-controlled robot. Our controller leverages a state-of-the art optimal control solver, namely Differential Dynamic Programming (DDP), in order to replan state and control trajectories at real-time rates (1kHz). In addition to this experimental proof of concept, we present exhaustive performance analysis on the iconic pick-and-place task and show that our controller outperforms open-loop MPC. We also exhibit the importance of a sufficient preview horizon and full robot dynamics in the controller performance through comparisons with inverse dynamics and kinematic optimization.