We develop in detail, the classical magnetic monopoles of non-abelian Yang-Mills gauge theory and show how these monopoles, when analyzed using Gauss’ / Stokes’ theorem, appear to confine their gauge fields, and also, appear to be composite objects. Of course, baryons, which include the protons and neutrons at the heart of nuclear physics, also confine their gauge fields and are similarly-composite objects. This raises the question whether the magnetic monopoles of Yang-Mills theory are in some fashion related to the observed physical baryons. After developing inverse solutions for the non-abelian electric charge densities while carefully examining uniqueness and gauge fixing, we use these solutions together with Dirac theory to “populate” these classical monopoles with fermions. Applying the Fermi-Dirac-Pauli Exclusion Principle to these fermions forces the selection of a dimension-3 gauge group initially chosen to be SU(3). We then find that these non-abelian magnetic monopoles have the exact chromodynamic symmetries of baryons and interact via colored magnetic fields with the exact chromodynamic symmetries of mesons. We show that a required U(1) factor ensures that these monopoles are topologically stable, and also “flavors” these monopole as protons and neutrons. Because this exposition is classical, we also discuss the extent to which classical field theory can be used to effectively analyze baryons and confinement. We point out how a recursive aspect of the non-abelian electric charge solution may be used to perform an analytically-exact quantum path integration for Yang-Mills theory, proving the existence of a non-trivial quantum Yang–Mills theory on R4 for any simple gauge group G. Finally, we use the results of this path integration to develop four examples of the application of analytical non-linear quantum field theory, which includes a quantum field explanation of confinement, a fitting of the running QCD curve to the known empirical data within experimental error bars, and a careful review of single and double slit experiments.