  We explore correlator product states for the approximation of correlated
wavefunctions in arbitrary dimensions. We show that they encompass many
interesting states including Laughlin's quantum Hall wavefunction, Huse and
Elser's frustrated spin states, and Kitaev's toric code. We further establish
their relation to common families of variational wavefunctions, such as matrix
and tensor product states and resonating valence bond states. Calculations on
the Heisenberg and spinless Hubbard models show that correlator product states
capture both two-dimensional correlations (independent of system width) as well
as non-trivial fermionic correlations (without sign problems). In
one-dimensional simulations, correlator product states appear competitive with
matrix product states with a comparable number of variational parameters,
suggesting they may eventually provide a route to practically generalise the
density matrix renormalisation group to higher dimensions.
