Glyphosate (Roundup, Eraser, Killzall and other brands) or triclopyr (Brush-B-Gon, Brush Killer, Cut Vine and Stump Killer and other brands) are commonly recommended for weedy vine control.
What kills vines permanently?
Glyphosate is best for herbaceous vines. The more serious commercial mixes with 2,4-D and Banvel can eradicate the vine but are more dangerous to work with and can possibly harm the soil. Herbicides are non-selective, so cover any healthy plants you want to keep with a tarp before you apply it.
How do I permanently get rid of bindweed?
As bindweed is a perennial weed, it can only be completely killed with the systemic weedkiller glyphosate. This needs to be applied to the leaves, which is then taken down into the roots as bindweed grows. Other types of weedkiller will kill only the top growth, and bindweed simply regrows from the roots.