Let M be a compact manifold with (or without) boundary and let \(f: R\times M\to TM\) (the tangent bundle of M) be a time-dependent vector field, periodic of period T with respect to the time parameter t. The authors consider the differential equation on M: \(\dot x(t)=\lambda f(t,x(t))\) (\(t\in R)\) for each parameter \(\lambda \in R\). By making use of the theory of fixed point index of continuous maps, similar to the classical Leray-Schauder index, the authors give some sufficient conditions for the above differential equation to have T-periodic orbits on M for each \(\lambda \in R\).