Abstract: Quantum memories have been realized in different physical systems, such as atomic ensembles or solid systems. To date, all quantum memories have realized the storage and retrieval of photons encoded using only a two-dimensional space spanned, for example, by the orthogonal polarization states, and hence can only store a quantum bit. Using electromagnetically induced transparency in a cold atomic ensemble, we report the experimental realization of a quantum memory storing a photon encoded in a three-dimensional space spanned by orbital angular momentum (OAM) states. We experimentally reconstruct the storage process density matrix with a fidelity of 85.3% +/- 1.8% using a 4-f imaging system. We also realize storage of two special photonic qutrit states as examples. Toward storing a higher-dimensional state encoded in OAM space, the efficiency difference between different OAM states should be considered according to the experimental results. The capability to store high-dimensional quantum states with high fidelity is a key step towards building high-dimensional quantum networks.