This is an introduction to the mathematical theory of the Yang-Mills equation, with an emphasis on its applications to low-dimensional topology. The course consists of two parts. In the first half, we develop the analysis of the ASD equation on four-manifolds and prove Donaldson's diagonalization theorem. Along the way, we introduce tools that are also useful in other fields of mathematics, such as principal bundles, spin structures, and the Atiyah-Singer index theorem. In the second half, we develop the singular instanton Floer homology theory and discuss its applications to knots and links.