Attention is attracted to the idea of using magnetic flux trapping (MFT) HTSC as a memory medium (MM). From this point of view the writing-storage-reading-erasing' cycle was studied. Investigations of the feasibility in principle of local writing and measurements of local MFT dependences on temperature, external magnetic field and transport current have been performed. For YBa₂Cu₃O_y ceramics the following values of local trapped magnetic fields were obtained: H_lt approximately=20-30 Oe at 77K; H_lt approximately=60 Oe at 4.2 K, for YBa₂Cu₃O_y films at 4-9 K, H_lt>or approximately=100 Oe. From the local MFT relaxation law it follows that for a time period of 10 to 10⁵ h, the H_t drop does not exceed 10%. The 'writing-erasing' cycle was reproduced several hundred times at least. The necessity of optimum HTSC technology development for MM is emphasised. Prospective HTSC applications as MM as well as memory elements are discussed.