  A general procedure to reveal an Abelian structure of Yang-Mills theories by
means of a (nonlocal) change of variables, rather than by gauge fixing, in the
space of connections is proposed. The Abelian gauge group is isomorphic to the
maximal Abelian subgroup of the Yang-Mills gauge group, but not its subgroup. A
Maxwell field of the Abelian theory contains topological degrees of freedom of
original Yang-Mills fields which generate monopole-like and flux-like defects
upon an Abelian projection. 't Hooft's conjecture that ``monopole'' dynamics is
projection independent is proved for a special class of Abelian projections. A
partial duality and a dynamical regime in which the theory may have massive
excitations being knot-like solitons are discussed.
