Invasive plants are those that multiply rapidly and overwhelm an area. They crowd out and smother other plants and clog wetlands and waterways. They are usually non-native plants that have escaped cultivation and spread rapidly in Florida's climate.
Brazilian pepper tree 
favored by several Florida butterflies and birds is one of the most invasive plants in Florida. The seeds, which germinate in most ecosystems, are widely distributed by birds. It is now prohibited and should be destroyed.
Skunk vine (
Paederia foetida)

is an aggressive plant found throughout the Southeast. It is running rampant, particularly in central Florida where it smothers native plants -- and everything else -- and provides a "fire ladder" to tree tops.
The vines often reach 30 feet in length -- which is misleading, they grow 30 feet and their offshoots grow 30 feet and their offshoots grow 30 feet and so on. It often reaches 30 feet in length quickly. It grows anywhere, even underwater.
You probably have skunk vine if:
- Something is twining all through your plants.
- When you grab it, it smells like rotten potatoes.
- When you pull it, it stretches.
- When you pull harder, it stretches, snaps, and you have a tiny piece in your hand.
- When you finally get a good grip on it, you find that it's snaked through grass and plants in all directions.
- You come back from vacation and cannot find your hedge or your lawn or both.
- Where you once had a tree, you now have a large lump of green stuff.
- You realize I'm not kidding.
If you're plagued with this stuff, you'll go through stages: fiercely attacking it, admiring your progress, fiercely attacking it, admiring your progress, fiercely attacking it, grumbling in frustration, fiercely attacking it, grumbling more and louder, fiercely attacking it, astonished that you could feel such vehemence against a plant, fiercely attacking it, wanting to cry.
Center for Aquatic and Invasive Plants:
Skunk Vine
IFAS:
Natural Area Weeds: Skunk Vine
There are controls. They're herbicides. They require covering the foliage. That will kill what the skunk vine is growing on. Is skunk vine serious enough to resort to chemical controls? At a cost of more than $1500/acre? Does it matter if we kill everything to kill the skunk vine when it's going to kill everything anyway?
But there
is hope...
Pemberton and Pratt.
They're entomologists at the Invasive Plant Research Laboratory in Fort Lauderdale who are looking for a way to control skunk vine. That's bio control they're looking for, not chemicals, particularly insects that have an appetite for only skunk vine and not for any other plants. Sounds unlikely? You know how picky some butterflies are. They have found some candidates. Keep your fingers crossed.
An introduction to Robert W. Pemberton and Paul D. Pratt:
Scouring the World for a Skunk Vine Control.
For more details, check out
Biological Control of Skunk Vine by Pemberton and Pratt.
Other plants that should be destroyed include
kudzu (
Pueraria lobata) which overwhelms pine forests in North Florida,
Australian pines (
Casuarina spp.), and punk trees (
Melaleuca quinquenervia). The latter two are primarily a problem in South Florida.
The pines spread by seeds and suckers. Their dense foliage crowds out native trees and shrubs and fallen foliage smothers other plants and releases a chemical that suppresses growth of other vegetation. The pines also have shallow roots and are prone to toppling in high winds. The
Melaleuca, also called the cajeput or bottle brush tree, has invaded the Everglades, crowding out other species. Many people are allergic to it and it can also be a fire hazard.
Some native plants also have the potential to become invasive given the proper conditions. If you elect to grow these plants, please take steps to prevent their spread.
The
Florida Exotic Pest Plant Council site has a listing of invasive plants and information about their control. The University of Florida's
Center for Aquatic and Invasive Plants provides a wealth of information, including photos and identification guides.
The
Tampa Bay Estuary Program offers a
A Field Guide to Invasive Plants of the Tampa Bay Region in PDF format.