A manifold includes a tangent space associated with each point. A frame defines a basis for the tangent space at each point, and a connection allows us to compare vectors at different points, leading to concepts including the covariant derivative and curvature. All of these concepts can be applied to an arbitrary vector space associated with each point in place of the tangent space. This is the idea behind gauge theories. Both manifolds with connection and gauge theories can then be described using the mathematical language of fiber bundles.