Mints can be a BIG problem, Sharon.
You have two non-herbicidal methods to get rid of your invasive mint: digging it out, or smothering it.
If you have other established plants, dig them out, remove every piece of mint stem and rhizome from their root balls, and relocate them. Then dig the area deeply and thoroughly, shaking mint stems and roots from each spade of soil. Some people recommend screening each spadeful of soil. Let the area sit for a few weeks, digging or pulling any new mint plants that emerge (they will!).
Smothering will take a full season. After you've relocated any other plants you want to save, spread several overlapping layers of cardboard or a thick layer of newspaper over the entire area. Don't leave any gaps that will let the light shine through. Then cover the paper with a thick layer of leaves, straw, hay, or pine needles. If it's an area well-suited to vegetable production, you could stab small holes in the mulch, plant winter or summer squash, pumpkins, or cucumbers and let them ramble over the ground as your mint is smothering underneath.The following spring, pull or dig any mints that emerge (some will). You can either remove the mulch, or add more and plant new crops.
Best method of all: dig the mint out first, then apply the smothering mulch.
If you do enjoy having mints in your garden, try planting them in containers (submerged in soil if you like), so the rhizomes can't encroach on neighboring plants. Check the containers for cracks before you sink them into soil.
great question. I hate this, especially when it gets under the deck. about 11 years ago.