The goal of this thesis was to mathematically explain concepts from gauge theories. The most emphasis was put on explaining gauge transformations, covariant derivatives, and curvature. The main tools we used for such description were principal fiber bundles and connections. As there exists a lot of background geometry, preparations had to be made before the introduction of those two. Those preparations included an understanding of fiber bundles and Lie groups. Fiber bundles allowed for the description of spaces are not trivial, i.e. those that can be described only locally. Lie groups, on the other hand, were used as a tool for the description of the symmetries we were dealing with. On principal fiber bundles connections were defined as a formal way of comparison that is not well defined in Euclidean geometry. This helped us understand gauge transformations. However, for a full description of covariant derivatives, associated bundles need to be mentioned as they serve as a tool for defining parallel transport map in the context of a vector bundle in which we know how to compare points. This is enough for a formal definition of a covariant derivative, which can be found in both particle physics and general relativity. Curvature relied on no more new concepts as understanding covariant derivative and exterior covariant differentiation was enough for its description. Examples of curvature in gauge theories were given alongside terminology used in these fields as naming conventions differ.