Quantum logic gates are the crucial information-processing operation of quantumcomputers. Two crucial performance metrics for logic gates are their precision andspeed. Quantum processors based on trapped ions have always been the touchstonefor gate precision, but have suffered from slow speed relative to other quantum logicplatforms such as solid state systems. This thesis shows that it is possible to acceleratethe logic "clock speed" from kHz to MHz speeds, whilst maintaining a precision of99.8%. This is almost as high as the world record for conventional trapped-ion gates,but more than 20 times faster. It also demonstrates entanglement generation in atime (480ns) shorter than the natural timescale of the ions' motion in the trap, whichstarts to probe an interesting new regime of ion trap physics. In separate experiments, some of the first "mixed-species" quantum logic gates areperformed, both between two different elements, and between different isotopes.The mixed-isotope gate is used to make the first test of the quantum-mechanical Bellinequality between two different species of isolated atoms.