Laser-cooled and trapped atomic ions form an ideal standard for the simulation of interacting quantum spin models. Effective spins are represented by appropriate internal energy levels within each ion, and the spins can be measured with near-perfect efficiency using state-dependent fluorescence techniques. By applying optical fields that exert optical dipole forces on the ions, their Coulomb interaction can be modulated to give rise to long-range and tunable spin-spin interactions that can be reconfigured by shaping the spectrum and pattern of the laser fields. Here we review the theory behind this system, recent experimental data on the adiabatic prepration of complex ground states and dynamical studies with small collections of ions, and speculate on the near future when the system becomes so complex that its behavior cannot be modeled with conventional computers.