Quantum many-body systems with tens or hundreds of particles are often intractable to simulate on classical computers due to the exponential growth of the Hilbert space with the number of particles. We can nonetheless hope to simulate them efficiently by using another quantum system'one that also shows this exponential scaling but is initializable, well controlled, and easy to manipulate and probe. We use a collection of trapped atomic ions as a platform for the quantum simulation of lattice spin models. The use of mode-locked lasers to create the state-dependent potentials that simulate a variety of spin Hamiltonians allows operation in a spectral region that is relatively free of laser-induced decoherence. We also realize ultrafast gates where a single laser pulse can drive a high-fidelity single-qubit gate in ~50 ps. Single laser pulses may also be used for rapid deceleration of (neutral) molecular beams, which may provide us with a bridge over the "temperature gap" for direct cooling of molecules to ultracold temperatures.