This paper is concerned with the propagation of traveling wave solutions for diffusive N-species Lotka-Volterra competition systems. We first establish an innovative lemma relating to the existence of positive solutions for the transpose systems of linear systems. Then a necessary and sufficient condition is established for the existence of non-decreasing traveling wave solutions connecting two different equilibria. In addition, using the two-sided Laplace transform, we can obtain the asymptotic behavior of traveling wave solutions at positive infinity. Based on the properties of asymptotic behavior, we show that all non-critical traveling wave solutions with the same wave speed are unique up to translations. We also provide an example to support our result.