Libertarianism is a school of thought that says humans are free from physical determinism and all the other diverse forms of determinism.
Libertarians believe that strict determinism and freedom are incompatible.
Most libertarians have been mind/body dualists who explained human freedom by a separate mind substance that somehow manages to act in the physical world. Some believed that our freedom only existed in a transcendental or noumenal world, leaving the physical world to be completely deterministic.
Religious libertarians say that God has given man a gift of freedom, but at the same time that God's foreknowledge knows everything that man will do.
In recent free will debates, it is maintained that libertarian accounts of free will are unintelligible. No coherent idea can be provided for indeterminism and chance, they say. 1
The first libertarian, Epicurus, argued that as atoms moved through the void, there were occasions when they would "swerve" from their otherwise determined paths, thus initiating new causal chains.
The modern equivalent of the Epicurean swerve of atoms is quantum mechanical uncertainty, again a property of atoms. We now know that atoms do not just occasionally swerve, they move unpredictably whenever they are in close contact with other atoms.
Everything in the material universe is made of atoms in unstoppable perpetual motion. Deterministic paths are only the case for very large objects, where the statistical laws of atomic physics average to become nearly certain dynamical laws for billiard balls and planets.
Many determinists are now willing to admit that there is real indeterminism in the universe. 2,3 Libertarians should agree with them that if indeterministic chance was the direct cause of our actions, that would not be freedom with responsibility.
Then determinists should agree that if chance was not a direct cause of our actions, it would do no harm. In which case, libertarians should be able to convince the determinists that if chance provides real alternatives to be considered by the adequately determined will, it provides real alternative possibilities for thought and action. It provides freedom and creativity.
Libertarians should give the determinists, at least the compatibilists, the kind of freedom they say they want, one that provides an adequately determined will for which we can take responsibility.
For Teachers
Clarke, R. (2003). Libertarian Accounts of Free Will, Oxford University Press.
Dennett, D. C. (1978). Brainstorms : philosophical essays on mind and psychology. Montgomery, Vt., Bradford Books. (see "Giving the Libertarians What They Say They Want.")
Kane, R. (2001). The Oxford handbook of free will. Oxford ; New York, Oxford University Press.
For Scholars

1. Clarke, Randolph (2003), Libertarian Accounts of Free Will, p.xiii.

Accounts of free will purport to tell us what is required if we are to be free agents, individuals who, at least sometimes when we act, act freely. Libertarian accounts, of course, include a requirement of indeterminism of one sort or another somewhere in the processes leading to free actions. But while proponents of such views take determinism to preclude free will, indeterminism is widely held to be no more hospitable. An undetermined action, it is said would be random or arbitrary. It could not be rational or rationally explicable. The agent would lack control over her behavior. At best, indeterminism in the processes leading to our actions would be superfluous, adding nothing of value even if it did not detract from what we want.

2. Honderich, Ted (2002), How Free Are You?, p.5.

"Maybe it should have been called determinism-where-it-matters. It allows that there is or may be some indeterminism but only at what is called the micro-level of our existence, the level of the small particles of our bodies."

3. Searle, John (2004), Freedom and Neurobiology, p.74-75.

"First we know that our experiences of free action contain both indeterminism and rationality...Second we know that quantum indeterminacy is the only form of indeterminism that is indisputably established as a fact of follows that quantum mechanics must enter into the explanation of consciousness."

Chapter 3.7 - The Ergod Chapter 4.2 - The History of Free Will
Part Three - Value Part Five - Problems