With　the　help　of　a　continuation　theorem　based　on　Gaines　and　Mawhin＇s　coincidence　degree,　easily　verifiable　criteria　are　established　for　the　global　existence　of　positive　periodic　solutions　of　a　delayed　ratio-dependent　predator-prey　system　with　stage　structure　for　predator.　The　approach　involves　some　new　technique　of　priori　estimate.　For　the　system　without　delay,　by　constructing　a　suitable　Lyapunov　function,　some　sufficient　conditions　which　guarantee　the　existence　of　a　unique　global　attractive　positive　periodic　solution　are　obtained.　Those　results　have　further　applications　in　population　dynamics.　Copyright　©　2005　Hindawi　Publishing　Corporation.