We describe correlator product states, a class of numerically efficient many-body wave functions to describe strongly correlated wave functions in any dimension. Correlator product states introduce direct correlations between physical degrees of freedom in a simple way, yet provide the flexibility to describe a wide variety of systems. We show that many interesting wave functions can be mapped exactly onto correlator product states, including Laughlin's quantum Hall wave function, Kitaev's toric code states, and Huse and Elser's frustrated spin states. We also outline the relationship between correlator product states and other common families of variational wave functions such as matrix product states, tensor product states, and resonating valence-bond states. Variational calculations for the Heisenberg and spinless Hubbard models demonstrate the promise of correlator product states for describing both two-dimensional and fermion correlations. Even in one-dimensional systems, correlator product states are competitive with matrix product states for a fixed number of variational parameters. © 2009 The American Physical Society.