At this point I usually like to remind people what they actually mean by determinism, and to point out that the concept isn't so easy.
Determinism is usually understood to be the principle by which "if we know everything about the current state of the universe, and we know the correct laws of physics, then we can determine uniquely what will be the future state of the universe".
In fact, of a given theory, it is pretty simple to find out whether it is deterministic or not: does the theory in question allow you to calculate what is the future state of a universe that follows exactly the theory, if you know whatever is to be (according to the theory) the current state, and the laws the theory stands for ?
Typically, Newton's mechanics is considered to be deterministic, up to one caveat: the fact that the "current state of the universe" in Newtonian mechanics is given by real numbers (positions and momenta), and that we don't know how to even write down the vast majority of real numbers (let alone "measure" it). This caveat is what gives rise to the field of "deterministic chaos". But apart from that, it is deterministic in that knowing current positions and momenta of all matter points and the correct force laws, allows you to know in principle the positions and momenta of those matter points in the future.
So there is already a caveat in Newton's mechanics itself (deterministic chaos), but moreover, the universe described by Newton's laws looks only approximately to the "real" universe.
The same can be said about classical electromagnetism (Maxwell's laws) and about classical relativity.
As others pointed out, quantum theory killed the idea. Now, there are interpretations of quantum theory which would make nature appear again as "deterministic", but where there is some property of nature which is part of the theory itself which doesn't allow any being of KNOWING the precise state of the universe at a certain moment (I'm thinking of Bohmian mechanics here). This means that even though nature might "be" deterministic, we cannot "exploit" that property because of some principle, which will forbid us to "know" the current state beyond a probabilistic description, and this probabilistic description will then propagate to the end result, only allowing for probabilistic predictions - as does standard quantum theory.
But I always like to throw in the following. When we talk about "laws of nature", usually we think about relatively succinct mathematical formulations: relatively "simple" mathematical structures that generalize easily to the situation at hand. But this doesn't need to be so. You could think of another kind of law of nature, one in which all past and future events are simply summed up in some kind of table. Call it "the Book". Now, we don't have "initial conditions" and a rather simple calculation rule on how to deduce future events, we simply have the full list of all events in the universe, as a "law of nature". There's of course no way to DISCOVER this "law of nature", but it is conceivable as law of nature. In fact, it is rather more amazing that extreme simplifications are possible, which reduce to just a few simple mathematical structures, applicable to a vast array of conditions.
So, in Newton's world, instead of giving you a simple differential equation from which we can DEDUCE all positions of matter points as a function of time, where this differential equation is the "law of nature", the law of nature in "the Book" would then simply be the functions of time which give us the positions of the matter points themselves. These functions are of course way more involved than just the differential equation to which they appear to be a solution, but that's just a matter of complexity of the laws of nature. What is amazing in physics, is that there are such immense "short cuts", such as a differential equation.
But let us for a moment imagine that the "true" laws of nature are just the "list of all events" (the Book). Now, is that "deterministic" or not ? On one hand, of course it is! If ever you had the Book, you could just READ what's going to happen, the future events are specified without any ambiguity. So it is of course deterministic. But there doesn't need to be a "law" that allows you from just the slice of "now" events to deduce the next slice of events. There may be only a statistical correlation if you insist on a "simple" mathematical law (a kind of correlation that's given by a theory like quantum theory for instance).
Or we could have a totally "random" universe, with no correlation between different slices of events. No causality at all, not even a probabilistic causality. This would be a very weird place to live in and certainly it is not our universe. But it is conceivable and it is even conceivable to have such a totally random universe, described by a very precise "Book", one in which every event is listed, but in which there are no obvious "simple" correlations between events in different time slices.
