A Memory Core is the part of an individual's mind that contains all of their memories. If a memory is removed from a Memory Core, it will be forgotten. It is possible to "enter" the bubbles to see the memory of the person and to remove the memory from the core to the real world. Once in the real world, it becomes a real object and can be destroyed easily.
In "Memory of a Memory", Ash tricks Finn and Jake into retrieving Marceline's memory of her breakup with Ash from her Memory Core, causing her to forget the event.
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Each memory seems to picture an object within a sphere from that specific memory, such as how the cherry blossoms relate to Marceline and Ash's breakup. In Marceline's memory core, pictures of the memories included cherry blossoms, a five-fingered hand, a wolf, a snowman, a shovel, a grave, a tree, an aquatic creature, and a candle.