Historically, in physics, hidden variable theories were espoused by a minority of physicists who argued that the statistical nature of quantum mechanics indicated that quantum mechanics is "incomplete". Albert Einstein, the most famous proponent of hidden variables, insisted that, "I am convinced God does not play dice — meaning that he believed that physical theories must be deterministic to be complete. Later, Bell's theorem would prove (in the opinion of most physicists and contrary to Einstein's assertion) that local hidden variables are impossible. It was thought that if hidden variables exist, new physical phenomena beyond quantum mechanics are needed to explain the universe as we know it.
The most famous such theory (because it gives the same answers as quantum mechanics, thus invalidating the famous theorem by von Neumann that no hidden variable theory reproducing the statistical predictions of QM is possible) is that of David Bohm, also known as the Causal Interpretation of quantum mechanics. Bohm's (nonlocal) hidden variable is called the quantum potential. Nowadays Bohm's theory is considered to be one of many interpretations of quantum mechanics which give a philosophical or realist interpretation, and not merely a positivistic one, to quantum-mechanical calculations. It is in fact just a reformulation of conventional quantum mechanics obtained by rearranging the equations and renaming the variables. Nevertheless it is a hidden variable theory.
The major reference for Bohm's theory today is his posthumous book with Basil Hiley.
In other words, quantum mechanics as it stands might be an incomplete description of reality. Some physicists maintain that underlying the probabilistic nature of the universe is an objective foundation/property — the hidden variable. Others, however, believe that there is no deeper reality in quantum mechanics — experiments have shown a vast class of hidden variable theories to be incompatible with observations.
Although determinism was initially a major motivation for physicists looking for hidden variable theories, nondeterministic theories trying to explain what the supposed reality underlying the quantum mechanics formalism looks like are also considered hidden variable theories; for example Edward Nelson's stochastic mechanics.
Physicists such as Alain Aspect and Paul Kwiat have performed experiments that have found violations of these inequalities up to 242 standard deviations(excellent scientific certainty). This rules out local hidden variable theories, but does not rule out non-local ones (which would refute quantum entanglement). Theoretically, there could be experimental problems that affect the validity of the experimental findings.
What Bohm did, on the basis of an idea of Louis de Broglie, was to posit both the quantum particle, e.g. an electron, and a hidden 'guiding wave' that governs its motion. Thus, in this theory electrons are quite clearly particles. When you perform a double-slit experiment (see wave-particle duality), they go through one slit rather than the other. However, their choice of slit is not random but is governed by the guiding wave, resulting in the wave pattern that is observed.
Such a view does not contradict the idea of local events that is used in both classical atomism and relativity theory as Bohm's theory (and indeed quantum mechanics, with which it is exactly equivalent) are still locally causal but allow nonlocal correlations (that is information travel is still restricted to the speed of light). It points to a view of a more holistic, mutually interpenetrating and interacting world. Indeed Bohm himself stressed the holistic aspect of quantum theory in his later years, when he became interested in the ideas of Jiddu Krishnamurti. The Bohm interpretation (as well as others) has also been the basis of some books which attempt to connect physics with Eastern mysticism and consciousness. Nevertheless this nonlocality is seen as a weakness of Bohm's theory by some physicists.
Another possible weakness of Bohm's theory is that some feel that it looks contrived. It was deliberately designed to give predictions which are in all details identical to conventional quantum mechanics. Bohm's aim was not to make a serious counterproposal but simply to demonstrate that hidden-variable theories are indeed possible. His hope was that this could lead to new insights and experiments that would lead beyond the current quantum theories.
Another type of deterministic theory was recently introduced by Gerard 't Hooft. This theory is motivated by the problems that are encountered when one tries to formulate a unified theory of quantum gravity.