I review recent work on realist Lorentz invariant formulations of
quantum theory. These take as ingredients a suitable initial state and
Hamiltonian or evolution law, and allow us to define a sample space of
configurations of beables and a probability distribution on that sample
space. It is then natural to postulate that physical reality corresponds
to one specific configuration, randomly chosen from the probability
distribution. I also explain how these ideas allow testable
generalizations of quantum theory.