I have had my fence for 30 years. The neighbor who moved in at the same time gave me permission to put the fence a few inches on on his property. The neighbor moved out about 15 years ago and is now demised. There have been several owners since then. No complaints. Now there is a new owner and they want me to move my fence. I live in Florida. Can they does this. Is this now my property?

Unfortunately I don't think you have a case for claiming this as your property. If the original neighbor had given you permission in writing, and if you had that recorded that writing then you would have an easement and the new neighbor could not force you to move the fence.

Since they original neighbor gave you permission you do not qualify to own this property by adverse possession.

If you don't want to move the fence then maybe the new neighbor would be willing to sell you an easement for the fence that would cost less than the cost of moving the fence.

It appears and I agree with Lisa, that given the way it happened, and not in writing, you will have to take the fence down. If not, they may do it and bill you for the cost.

At 15 years, perhaps time for new fence anyway. Are they willing to help with cost of new fence.

I agree with the others - you will have to move the fence. If you're wondering whether you could claim the land under adverse possession the answer is no. In Florida to claim adverse possession you must have open and notorious use of the land, meaning you do not allow any other person to use the land (i.e. it's fenced) and either (a) adverse possession under color of title - meaning that there's an error in your deed so it says your property includes this sliver of land, or (b) adverse possession without color of title, which requires you to have filed a return with the county appraisers within one year of gaining use of the property, and paying all taxes and liens assessed during possession of the property. I assume that your neighbors all along have been paying property taxes on the bit that's on your side of the fence, so this doesn't fit your case.

The easement like lisa said sounds best to me if they agree

... If the original neighbor had given you permission in writing, and if you had that recorded that writing then you would have an easement and the new neighbor could not force you to move the fence.
...

It's a moot point, because that's not the situation, but I believe "permission in writing" would have (depending of course on the language) been a revocable license, not an easement. So it wouldn't have helped.

It's a moot point, because that's not the situation, but I believe "permission in writing" would have (depending of course on the language) been a revocable license, not an easement. So it wouldn't have helped.

Of course it would depend on how it was written.