  The control of internal and motional quantum degrees of freedom of laser
cooled trapped ions has been subject to intense theoretical and experimental
research for about three decades. In the realm of quantum information science
the ability to deterministically prepare and measure quantum states of trapped
ions is unprecedented. This expertise may be employed to investigate physical
models conceived to describe systems that are not directly accessible for
experimental investigations. Here, we give an overview of current theoretical
proposals and experiments for such quantum simulations with trapped ions. This
includes various spin models (e.g., the quantum transverse Ising model, or a
neural network), the Bose-Hubbard Hamiltonian, the Frenkel-Kontorova model, and
quantum fields and relativistic effects.
