The authors consider the following delayed ratio-dependent predator-prey system
where and represent the predator and prey densities, respectively, , , , , and are positive periodic continuous functions with period , is a positive real constant. is a measurable, -periodic, normalized function such that . By using the continuation theorem of the coincidence degree theory [see R. E. Gaines and J. L. Mawhin, Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics. 568. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)], the authors establish two main theorems on the existence of at least one positive -periodic solution of system when the functional response function is monotonic or nonmonotonic. As corollaries, some applications are listed.