Coordinates are bridges between geometry and "numbers". If we want to assign coordinates to a family of objects parametrized by some space, we get a family of coordinate systems that depend on locations in the parameter space. This is pretty much what a gauge is. A gauge transformation is a change of coordinates applied to each such coordinate system in the family. A gauge theory is for systems to which gauge transformations can be applied, and very often we care about quantities that are invariant under gauge transformations. This talk will be focused on basic mathematical ideas behind classical gauge theory, illustrated with simple conceptual examples in geometry and combinatorics.