In this article a systematic study is given of the asymptotic behavior of two-sample tests based on U-Statistics with arbitrary antisymmetric kernels ψ. Besides the investigation under the hypothesis and under fixed alternatives we determine the local power as a function of ψ as well as its maximizing value ψ_opt. Moreover formulas for the asymptotic relative efficiency ARE(ψ₂, ψ₁) of the ψ₂-test with respect to the ψ₁-test are derived. It turns out that ψ_opt also yields the most efficient test in the sense that ARE(ψ_opt, ψ) ≤ 1 for all (admissible) kernels ψ.