A memory refreshing system that can dynamically execute a self-refresh operation, even though the computer system is in a normal operational mode and can thus save the power consumption, and includes a memory refreshing system, for a memory system including a plurality of memory banks, which comprises a memory refreshing device, provided in each of the memory banks, for performing a refresh operation within a corresponding memory bank by, e.g., units of rows in response to a refresh request; a first memory refresh controller for supplying a refresh request to the memory system every first time period; a second memory refresh controller, provided for each of the memory banks, for supplying a refresh request to the corresponding memory bank every second time period that is longer than the first time period; a memory access monitor, provided for each of the memory banks, for detecting an access to the corresponding memory bank; a timer, provided for each of the memory banks, for generating a timer output when a first predetermined time has elapsed since a last access to the corresponding memory bank, and for canceling a counted timer value in response to the next access to the corresponding memory bank; and a selector, provided for each of the memory banks, for selecting one of the refresh requests from the first and the second memory refresh control means in accordance with the timer output.