Determinism is the view that the state of the world at any moment, plus the laws of nature, determine (i.e., logically entail) the state of the world at any other moment. Quantum mechanics and chaos theory relate to determinism in rather different ways.
Chaos theory concerns systems whose development is exquisitely sensitive to their current state -- in that a very small change to their current state would produce enormous changes to their later state. A chaotic system is not incompatible with determinism as I have defined it above. But the existence of chaotic systems entails that any small uncertainty in our knowledge of that system's initial conditions (and some such uncertainty is always present, for grubby practical reasons) will quickly ramify into great uncertainty in our predictions regarding that system, even if we know all of the relevant laws of nature.
None of this threatens determinism as a view about prediction "in principle." But quantum mechanics does that. The complete state of a system, as quantum mechancs describes it, fails (when combined with the laws of nature given by quantum mechanics) to determine the outcomes of later measurements on that system. If quantum mechanics gives a complete description of the universe, then the universe is indeterministic.
Well, that's roughly right. It leaves out some important complications, such as (i) the fact that there are some deterministic interpretations of quantum mechanics, though those interpretations involve their own weird elements, and (ii) quantum mechanics is perfectly deterministic except insofar as it concerns measurement outcomes. That is, in a universe without observers but governed by quantum mechanics, determinism would hold. That observers (or measuring devices) have a special role in quantum mechanics is pretty weird -- arguably, a great deal weirder than indeterminism.