Quantum memories are indispensable for quantum communication and quantum computation, which are able to realize the storage and retrieval of a quantum state such as a single photon, entanglement, or a squeezed state on demand. Among those memories realized by different protocols, the Raman quantum memory has advantages in its broadband and high-speed properties, resulting in huge potential applications in quantum network and quantum computation. However, the realization of Raman quantum memory for a true single photon and photonic entanglement is a challenging job. In this review, after briefly introducing the main benchmarks for quantum memories, showing the state of the art, we focus on the review of the experimental progress recently achieved in storing the quantum state by Raman scheme in our group. We believe that all achievements reviewed are very hopeful in building up a high-speed quantum network.