Summary: The delayed Leslie-Gower predator-prey system is investigated. By choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of J. Wu
[Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034
)] for functional differential equations, we may show the global existence of periodic solutions.