This paper is devoted to study the well-known Razumikhin-type theorem for a class of stochastic functional differential equations with Levy noise and Markov switching. In comparison to the standard Gaussian noise, Levy noise and Markov switching make the analysis more difficult owing to the discontinuity of its sample paths. In this paper, we attempt to overcome this difficulty. By using the Razumikhin method and Lyapunov functions, we obtain several Razumikhin-type theorems to prove the pth moment exponential stability of the suggested system. Based on these results, we further discuss the pth moment exponential stability of stochastic delay differential equations with Levy noise and Markov switching. In particular, the results obtained in this paper improve and generalise some previous works given in the literature. Finally, an example is provided to illustrate the effectiveness of the theoretical results.