Local hidden variables have been ruled out, notably by experiments with Bell's inequality.
This means that quantum effects cannot be caused by some unseen property of the objects involved (e.g. if particles had some extra 'charge' we didn't know about).
The effects could be caused by some unseen "non-local" property, e.g. some property of space itself which propagates faster than light. A nice example is to model spacetime as a network (e.g. http://www.wolframscience.com/nks/p475--space-as-a-network ) with information/causality propagating along edges in the network. If most of the edges are short, and form a 3D lattice, we'd get the universe we're familiar with. We can then model quantum phenomena, like entanglement, by introducing some long edges, connecting regions of the network which would otherwise be far apart (the entangled particles).
Since there are so many possible non-local theories, with no way to distinguish between them, Physicists tend to prefer quantum descriptions of local, observable properties, since that seems like less of a leap of faith.
And if you assume standard probability theory holds. If you allow for so called Exotic Probability, then local hidden variable theories are not ruled out. See [1] for a list of relevant literature.
