1. Ivy 2. tree stumps
1. We assume you are referring to English ivy. This is a non-native invasive plant and not easy to eliminate, but very possible. If you have large expanses of it, mow it down as low as you can. Spray immediately with an herbicide that contains glyphosate. This is a systemic herbicide and travels through the system of the plant, down to the roots. As soon as you see any new growth (some spots will be missed, and the plants have a huge reserve of energy in their roots to help them regrow), spray or cut and spray again immediately. Do not let it regrow and build up its root system again.
The "Weeds Gone Wild" website has a great fact sheet for getting rid of English ivy: https://www.nps.gov/plants/alien/fact/hehe1.htm
When cutting ivy off of trees, you do not have to remove it all from the tree, just a band at the bottom. When you cut it at the base, treat the stubs with glyphosate.
Glyphosate is a total vegetation killer, so it must not contact your desirable plants. Use a shield of cardboard or plastic to protect nearby plants.
2. Trees (some more than others) also have energy reserves in their roots, which will enable them to resprout. Spray with glyphosate or simply remove the shoots as soon as they appear. Eventually this will exhaust the root reserves.
Another tip to prevent resprouting is to mangle the cuts. You can smash the cut with a tool, even a rock! This makes it harder for the plant to regrow from the same spot.