tl;dr: Is the world deterministic? Yes, no, yes, and no. Yes, because the universal state vector evolves deterministically. No, because of quantum randomness. Yes, because of determinism in classical physics. No, because of very limited information processing capacity of our brains.
Epistemic state: I've pondered about this for quite a while. By now, I've good confidence in each part.
A friend of mine once started a phone call with the question "Is the world deterministic?" The reasoning was that if the world is deterministic then we can't decide on what actions we take, that is, then there is no free will. And no free will meant an existential crisis. But let's back up first what I mean by "deterministic".
Determinism is the concept that any given state of a system has only one single successor state. A system is called deterministic, if it can evolve over time from its current state into only one pre-determined next state. So, now, is the world deterministic or not? I think there is a clear answer to that: yes, no, yes, and no. In that order. Let me explain.
For all I know, we have already found the equation that describes how the universe works. It's called the Schrödinger equation:
$$\dot \psi = -i\hat H \psi$$
In its most elegant form, it is a simple differential equation which describes how the state $\psi$ of the world evolves over time. While there are arguments, whether this is actually the final answer, we know for sure that any deeper mechanism must reproduce this equation. The crucial property I'd like to highlight is that this equation is a deterministic calculation. According to quantum theory the universe is deterministic. Therefore, yes, the world is deterministic.
Not so fast, experts in quantum physics among you might say now. Physical systems, as described by quantum mechanics, have some counterintuitive properties. One is that the state they deterministically evolve into usually describes a probabilistic distribution over multiple, mutually exclusive states (well, it's actually a bit more complex than that, but this description will suffice for now). Another is that whenever we look at a system, it seems to discontinuously collapse into one randomly selected, but very precise state. Hence, by any means, to have multiple follow-up states to one current state of the world is very possible, and the actually observed state is drawn randomly. Therefore, no, the world we can observe is clearly not deterministic.
Now, atoms are small. And they are many. REALLY many. In an average human body, there are in the order of $10^{28}$ atoms. At that scale, and observing that many systems show self-organizing behavior, the laws we would reasonably use are averages over very many particles. And expectation values, to which averages converge, behave deterministically, even in quantum physics. In other words, everything of interest in our everyday experience, even things too small to see with the naked eye, consists of enough elementary particles to behave almost deterministically. Therefore, for all that matters for us, the world is basically deterministic. Sorry to particle physicists.
On that level, there is still way too much we'd have to track to predict the world. Luckily, there are often higher-order emergent properties that we can use to predict the future to some degree. But because most of the information is hidden when using such approximations, these emergent properties can (and often do) still behave stochastically even though the underlying mechanism is deterministic. Indeed, chaotic systems (like the weather) quickly evade predictions even of the most sophisticated models computed on the largest supercomputers. Nevertheless, simple higher-order approximations are often accurate enough to allow us to make sense of the world, despite of the very limited and selective information that actually reaches our senses, and the very limited information processing capacities of our brains. We are probabilistic reasoners, weighing chances of what might happen next, and choose our next actions based on our best guesses. In our direct experience, the world is clearly not deterministic.
So, whether the world is deterministic or not depends on the domain you care about. If you look at the quantum level or everyday level, you'll likely fare better with probabilistic reasoning. If you're only interested in (large) sample averages, like computers or planetary motion, or ponder the fundamentals of reality, a deterministic view may be more appropriate.
Edit 21-02-23: Rewrote second last paragraph for more clarity.
Edit 09-07-24: Rewrote last paragraph.