Gravity is among the most ubiquitous forces within astrodynamics. The motion of every planet, asteroid, and spacecraft is intrinsically influenced by the gravitational forces of objects both near and far. Despite this, no universal model of this force exists. Rather, dynamicists must choose between many different gravity models that each carry their own unique advantages and drawbacks. For example, some models are fast to compute but lack analytic rigor; some achieve high accuracy but come with computational penalties, and others still are built with intrinsic assumptions or have limited operational validity. To combat these challenges, this thesis proposes the Physics-Informed Neural Network gravity model, or PINN-GM, which shifts attention away from analytic approaches and towards data-driven models. Specifically, the PINN-GM leverages recent advances in the field of Scientific Machine Learning to blend the power of neural networks with dynamical systems theory to produce high-fidelity models of complex dynamical systems without sacrificing analytic rigor. Through multiple iterations of development, the PINN-GM now offers high-accuracy, fast execution times, data efficiency, global validity, and exact differentiability. Taken together, these attributes make the PINN-GM well-suited to assist astrodynamicists in a variety of applications including reinforcement learning, periodic orbit discovery, and orbit determination.