The Yang-Mills existence and mass gap refers to a fundamental problem in theoretical physics concerning the behavior of certain gauge theories. These theories describe the interactions of elementary particles through force carriers, like gluons in quantum chromodynamics. The existence part of the problem asks whether a mathematically rigorous formulation of these theories can be established, while the mass gap aspect seeks to prove that there is a positive lower bound on the mass of particles in the theory. This problem is significant because it connects to the understanding of quantum field theory and the Standard Model of particle physics. A solution would not only confirm the theoretical predictions of particle masses but also enhance our understanding of the strong force that binds quarks together in hadrons. The mass gap implies that there are no massless particles in the spectrum, which is crucial for the stability of the vacuum in these theories.