[en] A comprehensive topological classification of vortices and their end-point Dirac monopoles is formulated in gauge theories with an arbitrary compact Lie group. By way of homotopy theory, a simple analysis is presented for the global groups U(1), O(3), and SU(2), then SU(N)/Z/sub N/ and SU(N). Finite vortices are achieved through the complementarity of Dirac strings and the nodal lines of the Higgs fields. In general, the varieties of topologically distinct vortices or monopoles are determined solely by the connectivity of the global group, specified by a discrete Abelian fundamental group. The close connection between our work and the Wu-Yang global formulation of gauge fields is pointed out