Chaotic dynamics of the Duffing oscillator with parametric and external forcing are studied. The unperturbed system, in which the damping and external forcing terms are removed and the parametric forcing amplitude or frequency is zero, has a hyperbolic saddle with a pair of separatrices. The following two cases are analyzed : (i) the parametric forcing amplitude is small ; and (ii) the parametic forcing amplitude is large but the frequency is small. The external forcing amplitude and damping coefficient are assumed to be small for both cases. Using a version of Melnikov's method, we obtain conditions under which the stable and unstable manifolds of a normally hyperbolic invariant torus intersect transversely and chaotic dynamics may occur near the unperturbed separatrices. Such chaotic motions are appropriately characterized by a generalization of the Bernoulli shift. We also describe the influence of parametric forcing on conditions for the occurrence of chaos. Numerical simulation results are given to demonstrate the theoretical results. Sustained chaotic motions wandering near the unperturbed homoclinic orbits are found.