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To exceed the borders or parameters of something. You're going to need a bigger pot, or else that soup will spill over. It's hard to contain violence to just one neighborhood, you know. It's only a matter of time until it spills over into our part of town too.
1. [for a container] to overflow. I hope your bucket of water doesn't spill over. The milk glass spilled over because it was filled too full.
2. [for the contents of a container] to overflow. The bucket is too full. I don't know why the water doesn't spill over. The milk spilled over because you overfilled the glass.