Abstract: Let $H(t)=-\Delta+V(t,x)$ be a time-dependent Schrödinger operator on $L^2(\R^n)$. We assume that $V(t,x)$ is $2\pi$-periodic in time and decays sufficiently rapidly in space. Let $U(t,0)$ be the associated propagator. We show that the point spectral subspace for the Floquet operator $U(2\pi, 0)$ is finite dimensional and that, for $u_0$ belonging to the continuous spectral subspace for $U(2\pi, 0)$, $U(t,0)u_0$ satisfies various dispersive properties: The solution $U(t,0)u_0$ has an asymptotic expansion in $t$ as $t\to\infty$ in the topology of $x$-weighted spaces; it satisfies the Strichartz estimate and the local smoothing property globally in time. We use the extended phase space formalism and the Floquet Hamiltonian for proving these properties.