We present a comprehensive geometric framework for gravity in which the fundamental dynamical variable is the distortion tensor encoding both torsion and non-metricity. After a thorough review of metric-affine geometry, we construct the most general action quadratic in the distortion and linear in the Levi-Civita curvature, supplemented by a Yang-Mills-type kinetic term. The theory propagates a massless graviton, a massive vector, and a massive scalar. We derive the full non-linear field equations, linearize them around Minkowski spacetime, perform a complete mode decomposition, and analyze the stability and unitarity of each sector. We study the low-energy limit and show that General Relativity is recovered exactly below the distortion mass scale. We then explore cosmological solutions, finding that the vector mode can drive late-time acceleration without a cosmological constant. Black hole solutions are analyzed perturbatively, revealing a possible mechanism for singularity regularization. We discuss gravitational wave signatures, including dipole radiation from the vector mode, and place preliminary constraints from LIGO/Virgo observations. Finally, we outline the power-counting renormalizability of the theory and its potential as an ultraviolet completion of General Relativity. Extensive appendices provide detailed derivations of all identities used in the main text.