We study what might be called fractional vortices, vortex configurations with the minimum winding from the viewpoint of their topological stability, but which are characterized by various notable substructures in the transverse energy distribution. The fractional vortices occur in diverse Abelian or non-Abelian generalizations of the Higgs model. The global and local features characterizing these are studied, and we identify the two crucial ingredients for their occurrence-the vacuum degeneracy leading to nontrivial vacuum moduli M, and the BPS nature of the vortices. Fractional vortices are further classified into two kinds. The first type of such vortices appear when M has orbifold Z(n) singularities; the second type occurs in systems in which the vacuum moduli space M possesses either a deformed geometry or some singularity. These general features are illustrated with several concrete models.