We review a class of efficient wavefunction approximations that are based around the limit of low entanglement. These wavefunctions, which go by such names as matrix product states and tensor network states, occupy a different region of Hilbert space from wavefunctions built around the Hartree–Fock limit. The best known class of low entanglement wavefunctions, the matrix product states, forms the variational space of the density matrix renormalization group algorithm. Because of their different structure to many other quantum chemistry wavefunctions, low entanglement approximations hold promise for problems conventionally considered hard in quantum chemistry, and in particular problems which have a multireference or strong correlation nature. In this review, we describe low entanglement wavefunctions at an introductory level, focusing on the main theoretical ideas. Topics covered include the theory of efficient wavefunction approximations, entanglement, matrix product states, and tensor network states including the tree tensor network, projected entangled pair states, and the multiscale entanglement renormalization ansatz. © 2012 John Wiley & Sons, Ltd.