[en] We introduce two maximal non-abelian gauge fixing conditions on the space of gauge orbits M for gauge theories over spaces with dimensions d ≤ 3. The gauge fixings are complete in the sense that they describe an open dense set M₀ of the space of gauge orbits M and select one and only one gauge field per gauge orbit in M₀. There are no Gribov copies or ambiguities in these gauges. M₀ is a contractible manifold with trivial topology. The set of gauge orbits which are not described by the gauge conditions M/M₀ is the boundary of M₀ and encodes all non-trivial topological properties of the space of gauge orbits. The gauge field configurations of this boundary M/M₀ can be explicitly identified with non-abelian monopoles and they are shown to play a very relevant role in the non-perturbative behavior of gauge theories in one, two and three space dimensions. It is conjectured that their role is also crucial for quark confinement in 3+1-dimensional gauge theories