But I think it's worth talking about what exactly we mean when we use the term "free will", and discussing implications of new advances in neuroscience that help us understand how we make our choices.
No sensible person would deny that our choices are heavily constrained by a number of factors. Brain damage can render us unable to comprehend choice, to empathize with others, or to think abstractly. Drugs can alter our moods and affect how we react to physical stimuli. We have no control over our moods, our environment, our subconscious thoughts, our memories, or which thoughts "just pop in there" – we can only react, and act volitionally strictly within the limited framework we are presented. Our "free will" is the product of a litany of causally determined processes that delineate and constrain our possible choices.
But so-called "hard determinists" (which I think is what most people associate with the word determinism) take this reasonable observation a step further, and opine that even our constrained choices are pre-determined; we're essentially automatons living in a universe whose every tick has been ordained since the moment of the Big Bang, and do nothing truly of our own volition. But this view of the universe is a dated one, rooted in a Newtonian view of reality. Isaac Newton's universe was like a clock, neatly unwinding as entropy increases. But Newton's view of the universe has been undone by quantum mechanics. Stephen Hawking, in The Grand Design, p. 72:
... the outcomes of physical processes cannot be predicted with certainty because they are not determined with certainty. Instead, given the initial state of a system, nature determines its state through a process that is fundamentally uncertain. In other words, nature does not dictate the outcome of any process or experiment, even in the simplest of situations. Rather, it allows for a number of different eventualities, each with a certain likelihood of being realized.But aside from physics, there may be perfectly reasonable biological reasons to believe we have free will. It's certainly true that we are composed of trillions of biological parts which each act in a deterministic, mechanistic fashion. But an important idea in biology is that the whole is not merely the sum of its parts. I've no qualms with acknowledging that consciousness arises as the product of mechanistic parts which act in a deterministic way – but the resulting whole is one which allows for a certain degree of volition within given parameters. That may not quite be the kind of mystical free will that dualists imagine, where consciousness is some ethereal force running on the brain like software; but this is still the kind of free will which allows us to, given our circumstances, make meaningful choices. Dualism doesn't actually give us answers so much as it pushes the question back a step into the realm of the ethereal and unfalsifiable.
What about neurology? Much has been made of the decision without awareness research which seems, at a glance, to bolster the notion that free will is an illusion: experimenters were able to use brain activity to predict subjects' decisions (to push one of two buttons) before they were aware of making them. But there are a few reasons for the determinists to hold off the parade: firstly, the predictions were not always correct; in fact, I looked up the study and learned they were only correct about 60% of the time, which doesn't seem particularly astounding to me. And, as others have noted, that study may reveal more about the cognitive mechanisms required for button-pressing rather than indicating the subversion of free will. But most importantly, this is a rudimentary task, and it tells us nothing about complex decision making in which we are weighing multiple possible outcomes.
I'm not advocating some kind of contra-causal free will which suggests that an ethereal force of consciousness must somehow circumvent the physical reality it inhabits. I think that's a bit absurd. And from a theological standpoint, it doesn't make much sense to me to suggest that God would design this marvelous perfect world, but then make it so that the function of consciousness required a perpetual circumvention of the laws he created. But I think that the concept of free will is perfectly compatible with both the fundamentally indeterminate nature of the universe (per Hawking), and with our evolutionary biology, which has imbued us with the ability to consider multiple possible outcomes or variables, and alter our actions accordingly.