Quantum-chemical calculations scale poorly with molecule size due to the complexity of simulating electronic and nuclear motion. Quantum computers could solve this problem by enabling exponentially faster simulations. Our recent results bring forward useful quantum computing for chemistry through orders-of-magnitude reductions in required quantum resources. We developed an approach for performing fully non-adiabatic simulations of chemical dynamics using trapped-ion quantum computers by mapping nuclear motion onto the motion of the trapped ions. Our experimental demonstrations have led to the first simulations of chemical reactions on a quantum computer, the best quantum-computer simulation of spectroscopy, and the first direct observation of geometric-phase interference in dynamics around a conical intersection. These demonstrations pave the way for near-term applications of existing quantum technology in computational chemistry as some of the first useful applications of quantum computing.