In 1954 C.N. Yang and R. Mills proposed a model for strong interactions in atomic nuclei. The main role in the classical version of the model was played by certain "physical fields" now called Yang-Mills fields. Mathematically, these were connections on certain vector (or principal) bundles which were supposed to satisfy a set of canonical PDEs (now Yang-Mills equations). The equations were Euler-Lagrange equations for the energy functional defined by the curvature of the connection. Almost three decades later mathematicians started to study solutions to such PDEs and got unexpected results.

We will give a gentle overwiew of results of Karen Uhlenbeck (Abel Prize 2019). These will include: existence and regularity of a connection given its curvature, solutions to Yang-Mills equations and their singularities, regulartity and singularities of harmonic maps. We will briefly mention how Uhlenbeck's results helped S. Donaldson to obtain his revolutionary results in topology of 4-manifolds. The gauge symmetry of the set of solutions to Yang-Mills PDEs was used for defining invariants of differentiable manifolds.