The problem of pth moment asymptotic stability analysis is considered for stochastic delayed hybrid systems with Levy noise. By virtue of Ito's formula and M-matrix theories, we propose some sufficient conditions to guarantee the asymptotic stability and exponential stability of the system. The criterion of mean square asymptotic stability is derived as well for delayed neural networks with Levy noise. A numerical example is provided to show the usefulness of the proposed asymptotic stability criterion.