Abstract We investigate the properties of Z( N) topological excitations in Wilson's lattice formulation of SU( N) Yang-Mills theories. We exhibit the Z( N) topological excitations as exact classical solutions on the lattice. After giving detailed qualitative discussions about the Z( N) excitations and their relevance to confinement, we investigate the Z( N) lattice gauge theories with the Wilson action and show that Z(2), Z(3) and Z(4) models are self-dual systems. (The self-duality of the Z(2) case has been known previously.) This property enables us to locate the critical points exactly in those systems under the assumption that the phase transition occurs at only one point in the coupling constant space. We then derive the effective action for the Z( N) topological excitations in the lattice SU( N) Yang-Mills theories in the steepest descent approximation. The critical coupling constants in the SU( N) models corresponding to the phase transition caused by the Z( N) excitations are estimated by using the information on the Z( N) models with the Wilson action. It is quite probable that the estimated value g r 2/4π 2 ∼ 1 31 (for SU(3)) is an upper bound. This indicates that the Wilson model of the SU(3) gauge field can be effective action of the QCD gluons which exhibit permanent quark confinement and, at the same time, freedom up to the distance characterized by the energy, at least, ∼1 TeV.