Following up the answer by Paul Siegel, in addition to Donaldson's spectacular result described by Paul, there are the papers of Sacks-Uhlenbeck on harmonic immersions of $2$-spheres into Riemannian manifolds and subsequent papers by Uhlenbeck on the "bubbling" phenomena for self-dual Yang-Mills connections and Taubes on gluing "bubbles" onto a self-dual Yang-Mills. These are the key technical results used by Donaldson in his thesis, as cited by Paul. They themselves laid the groundwork for a tremendous amount of work in geometric analysis since then.

Here are only a few that I recall offhand:

1. Minimal hypersurfaces (Schoen-Simon-Yau, Anderson)

2. Einstein manifolds (Gao, Anderson-Cheeger)

3. Recent work of Naber and Valtorta on stationary Yang-Mills connections

4. Recent work of Sung Jin Oh showing that similar phenomenon exists for the hyperbolic Yang-Mills equations