The chapter is devoted to the physics and the mathematics of Yang-Mills instantons. We first study a quantum mechanical toy model that in part teaches us under what circumstances non-perturbative saddles must be included in the path integral and how to implement these effectively. We then move on to explore Yang-Mills instantons and the saddle contributions thereof in great depth. The field strength 2-forms of Yang-Mills instantons obey (anti-)self-duality, whereby the Yang-Mills action saturates a topological lower bound. The solutions come with many deformation parameters, and the Atiyah-Singer index theorem is invoked to count these moduli for an arbitrary collection of (anti-)instantons in Yang-Mills theory with an arbitrary Lie algebra. In particular, we pay close attention to the question of minimal instantons in various Lie algebras and the resulting zero-mode countings, which separate out su(N) and sp(k) from the rest. Under the dilute instanton gas approximation, we observe how the sum over these topological saddles is captured by the insertion of effective instanton and anti-instanton vertices and how one-loop corrections thereof merely renormalize these vertices.