Gauge Flow Models are a new class of generative flow models that integrate a learnable Gauge Field into the Flow Ordinary Differential Equation (ODE). In this paper, we provide a comprehensive mathematical framework detailing the construction and properties of this model. Experimental results on Flow Matching for Gaussian Mixture Models demonstrate that Gauge Flow Models significantly outperform existing Flow Models of similar or larger size. Furthermore, unpublished research results suggest that it may provide improved performance on a wider range of generative tasks.