Most robotic behaviours focus on either manipulation or locomotion, where tasks that require the integration of both, such as full-body throwing, remain under-explored. Throwing with a robot involves complex coordination between object manipulation and legged locomotion, which is crucial for advanced real-world interactions. This work investigates the challenge of full-body throwing in robotic systems and highlights the advantages of utilising the robot's entire body. We propose a deep reinforcement learning (RL) approach that leverages the robot's body to enhance throwing performance through a strategically designed curriculum to avoid local optima and sparse but informative reward functions to improve policy flexibility. The robot's body learns to generate additional momentum and fine-tune the projectile release velocity. Our full-body method achieves on average 47% greater throwing distance and 34% greater throwing accuracy than the arm alone, across two robot morphologies - an armed quadruped and a humanoid. We also extend our method to optimise robot stability during throws. The learned policy effectively generalises throwing to targets at any 3D point in space within a specified range, which has not previously been achieved and does so with human-level throwing accuracy. We successfully transferred this approach from simulation to a real robot using sim2real techniques, demonstrating its practical viability.