We sometimes hear that science shows there’s no free will. New Scientist editor Graham Lawton argues: “Everything that has or will happen was determined at the big bang—and given that our brains are part of the physical universe, free will does not exist.”
Actually, quantum physics fatally undermines Lawton’s thesis by demonstrating that, at its most basic level, the universe is determined only statistically, not completely. And the brain, at its most basic level, is a quantum system.
In 2012, neuroscientist Matthew Lieberman admitted in Psychology Today that his own materialist view is an “unfalsifiable” leap of faith—true, of course, but not so honestly admitted in the past. Today, even the famous 1983 “Libet” experiment, once generally taken to demonstrate that there is no free will (because brain signals for action start before you are conscious of them), is now questioned in followup research. The neuroscience test environment is, of course, highly artificial. As Economist editor Anthony Gottlieb observes,

… looking at flickers of activity inside our heads can be a misleading way to see how our minds work. This is because many of the distinctively human things that people do take place over time and outside their craniums.

In Financial Times, Julian Baggini also points out, “we may not have as much conscious control over our actions as we think we do, but people who deny we have any at all have simply drawn the wrong lessons from neuroscience.” Neuropsychiatrist Jeffrey Schwartz, author of The Mind and the Brain, notes pragmatically that whether or not we have free will, we do have “free won’t”. In real-life situations, the ability to say no is often the quality that matters most.
The social outcomes of “no free will” are also problematic. John Monterosso and Barry Schwartz warn against brain-based interpretations of human behaviour in the New York Times (2012), pointing out that what people believe about free will influences their moral attitudes. In a study where they gave people either brain-based or mind-based information, “A brain characteristic that was even weakly associated with violence led people to exonerate the protagonist [person accused of a crime] more than a psychological factor that was strongly associated with violent acts.” In short, the “less free will” explanation reduced volunteers’ sense of moral responsibility.
Some hope to rescue free will while retaining materialism. Neuroscientist Eddy Nahmias argues, “A more complete scientific theory of the mind will have to explain how consciousness and rationality work, rather than explaining them away,” but he proposes to explain them in materialist terms. Neuroleadership Institute’s David Rock argues that because a belief in free will is useful, maybe we should be trying to get more people to believe in free will, without worrying about whether it is true.
Psychologist Roy F. Baumeister offers a more promising approach: “There is no need to insist that free will is some kind of magical violation of causality. Free will is just another kind of cause.”
Tellingly, he also offers a theory of how it evolved: “It’s what enables humans to control their actions in precisely the ways required to build and operate complex social systems.” True, but not informative.  What’s significant is that he offers an evolution-based explanation of why free will exists; not why it doesn’t.
And then there is the curious case of neuroscientist James Fallon. Reviewing brain scans of murderers, schizophrenics, and depressives, he came across one that was “obviously pathological.” It was his own. Later, he discovered he was distantly related to some murderers and that he had habits people didn’t like. He told the Smithsonian,

“I’m obnoxiously competitive. I won’t let my grandchildren win games…and I do jerky things that piss people off,” he reportedly said. “But while I’m aggressive...my aggression is sublimated. I’d rather beat someone in an argument than beat them up.”

Hmmm. A skeptic might note that statistically we are probably all distantly related to some murderers. And winning arguments is part of his conventional university job.
Fallon, who has written The Psychopath Inside,

previously believed one’s genetics predominately determined their life path, now has reportedly changed his tune on the issue.

“I was loved, and that protected me,” he told Smithsonian of the nurturing childhood he received from his doting parents, while also crediting the role free will plays in overcoming biology.

Maybe. The Catholic Church teaches that the more virtue one practices, the more free will one has. That approach also offers hope to people whose upbringing was more checkered than his. Denyse O'Leary is co-author of The Spiritual Brain. See also: “The Slow Death of a Pseudo-discipline” here.