Take a picture, use a cheap weed-whacker and throw the residue back over the fence.

Round up or the more powerful shrub and brush killers that are available at the big box home centers should do the trick. Spray a couple times a year and it should keep anything from growing. It might kill the whole plant even if you only spray on your side fo the fence but there is not much you can do to control that

You can dig a deep trench and use aluminum roof flashing as a root barrier. But why? Agapanthus are pretty. Iris are pretty. Let them take over and just remove the weeds and pretty soon they'll be so thick the weeds won't grow and you won't have any "gardening" to do there.

yes that's right--Agapanthus!!
It "would be nice with all of those........"but didn't get to choose.
the driveway along the fence is about 50' long I'm guessing till it hits the garage. The space between the driveway and fence is about 2'. The driveway itself is barely wide enough for a car (luckily I have a smaller size car.)
My landlord bought Nandina-the size that grows tall and will thicken up (I like it, plus not much water) and they are the entire length of the driveway. I had dug up the wild irises and a few Fotinia (hate those, I'm allergic to that yellow dust in the spring) so it looks nice now, but............those neighbor's plants come under. I was out there today putting down some extra dirt (since I shaved off a layer where the irises were then I put a double weed cloth barrier and then medium size redwood mulch bark. so hopefully that will help.
The Agapanthus bulbs that stick under the rotting fence are really mushy--I wonder if they are not healthy?? I chiseled as best I could then put the landscape fabric down up tight and upwards at the fench. but I think the roof flashing idea is great , I actually put a piece of roof tar style shingle at one spot.(it's just for 3 areas along this fence. ) BTW, do I cover the entire area with the bark, or do I leave an opening around each bush? (I did cut a nice hole in the fabric around each bush, plus it has the tubing drip system around each plant.)

Landscape fabric is a waste of time. The weed roots will grow into it from above from wind dropped seeds and it won't stop the more aggressive rhizomatous plants from tunneling through it from above. The Bermuda grass will win every time. The root barrier and a thick layer of mulch is all you need, along with monthly policing for any intruders that try to sneak their stems over the barrier.

"It might kill the whole plant even if you only spray on your side fo the fence but there is not much you can do to control that"

And bring complaints from the neighbor.

You can cut them back to the property line, but cannot kill them on the other side.

Just like trees.

Sorry, maybe I'm missing it, but why are you determined to make more work for yourself, for no good reason?

I have been an avid gardener for years. As I see it, these may be your best bets:

1. Shelve the attitude about your neighbor and go pay a visit. Their yard may not make the cover of House and Garden, but that doesn't mean they don't care about it just as much as you care about your space. People express their creativity in different ways and are dealing with who knows what these days; I'm sure you do stuff that others would mock, too.

Ask about the plants. Maybe they would be willing to move them. It's better than just killing something that isn't yours because you don't like it. And honestly, if you sprayed Round-Up and killed something of mine (especially if it had sentimental value), our nice neighborly relations would definitely be strained, to say the least. There's no reason to cultivate strife, so just go over and be nice. There's a chance they didn't even know their plants were encroaching.

2. As someone else suggested (another gardener, probably) plant the visitors so you can cut down on weeding that area. You know, what you consider a weed (agapanthus) is a plant that I've seen cultivated (and coveted) in various spaces around Monet's garden at Giverny and a plant that people like me pay about $9 each for in catalogs. One person's weed is another person's treasure. It's all in how we view things.

Irises are pretty, as are the "weeds" (which are probably mushy because they can't dry out, with being close to the dirt and bending under the fence). So why not create a garden out of them along the fenceline? You can get a few companion plants you like on the cheap, too. I bet it would be beautiful once you've finished.

Here is a link that might be useful: Agapanthus from a nursery catalog

I want to stop the neighbors ivy from traveling under my fence. I like the alum. roof flashing idea if that works. How deep below the surface of the ground does the flashing need to go? 6" or 12" maybe?