Author’s abstract: We obtain conditions for permanence (i.e. uniform persistence) in some diffusive Lotka-Volterra systems modeling three interacting species. Some of the results are based on the Hale-Waltman acyclicity theorem [see e.g. J. K. Hale and P. Waltman, SIAM J. Math. Anal. 20, No. 2, 388-395 (1989; Zbl 0692.34053)], others on average Lyapunov functions. All the results on permanence use hypotheses involving the signs of the principal eigenvalues of the associated linear elliptic operators.