We present several schemes for heralded storage of polarization states of single photons in single ions, using the 40 Ca + ion and photons at 854 nm wavelength as specific example. We compare the efficiencies of the schemes and the requirements for their implementation with respect to the preparation of the initial state of the ion, the absorption process and its analysis. These schemes may be used to create and herald entanglement of two distant ions through entanglement swapping; they are easily adapted to other atomic systems and wavelengths.The controlled transfer of quantum states between different systems is a cornerstone in the realization of quantum networks. One proposed implementation [1] of such a network employs single atoms for storing information (i. e., as nodes of the network) and single photons for transmitting information (i. e., as channels of the network); they are interfaced through controlled absorption and emission.In atom-photon interaction, the conservation of angular momentum relates the internal state of the atom with the polarization state of the exchanged photon. In emission, this generates atom-photon entanglement which may be used to prepare a certain superposition of Zeeman sublevels in the atom by measuring an emitted photon in a certain polarization basis [2,3], or to preserve such a superposition state in the context of quantum error correction [4]. In absorption, a certain polarization state of the photon excites the absorbing atom into the corresponding superposition of Zeeman sublevels. Using neutral rubidium-87 in this way as a quantum memory for an arbitrary polarization state of a photon at 780 nm has been proposed by [5] and put into effect by [6,7], using optical cavities to attain high efficiency.Here, we propose schemes employing both absorption and emission to implement heralded storage of the polarization state of a photon. First, the atom is prepared in a pure state ready to absorb a single photon. The absorption of a (certain but unknown) polarization state excites the atom into a corresponding superposition. Afterward, the excited atomic levels decay to different final levels, and another photon is emitted, which is entangled with the atom. By choosing the proper detection basis for this photon, the emission channels are made indistinguishable and the detection heralds the storage process but does not reveal the stored information. Due to the heralding, high fidelity will be achieved independent of the efficiency of the process. We describe in detail the requirements and conditions for such heralded quantum storage processes for the case of singly ionized calcium-40, the species used in our [8][9][10][11] and other experiments [12][13][14][15][16]. Application to other species is straightforward.In the proposed quantum-memory schemes, a single 40 Ca + ion is used to store the polarization state of a single photon at 854 nm wavelength (cf. fig. 1 a): the absorption happens on the 3 D 5/2 -4 P 3/2 transition and the emission of the herald takes place on the 4 P 3/2...