Spatiotemporal pulse shaping used chromatic aberration coupled with diffractive focusing to control the trajectory of a high-intensity laser spot and match the velocity of accelerating particles over distances significantly longer than the Rayleigh length. Simulations predicted the use of this chromatic flying focus to generate ionization waves at any velocity and experiments have demonstrated the ability to control their trajectory. The flying focus was proposed to overcome limitations in using plasmas to amplify ultrashort-pulse lasers, to frequency shift photons from the visible to the extreme ultraviolet over a few millimeters, and for nonlinear Thomson scattering. To maintain ultrashort pulse durations in the focal region, an achromatic spatiotemporal system was invented that takes advantage of spherical aberration to create an extended focal region and a novel radial echelon to control the velocity of the focal spot. Applying this achromatic flying focus to a laser-wakefield accelerator was shown to overcome dephasing where the ponderomotive force of the spatiotemporally shaped pulse drove a wakefield with a phase velocity equal to the speed of light in vacuum, preventing trapped electrons from outrunning the wake. Simulations in the linear regime and scaling laws in the bubble regime illustrate that this dephasingless LWFA can accelerate electrons to high energies in much shorter distances than a traditional LWFA—a single 4.5 m stage can accelerate electrons to TeV energies without the need for guiding structures. Furthermore, the flying focus was shown to vacuum accelerate electrons by reducing the velocity of the focal spot such that the ponderomotive force on the rising edge of the laser was sufficient to accelerate electrons beyond the speed of the subluminal flying focus. Theory and simulations demonstrate that these dynamic intensity peaks can backwards accelerate electrons to the MeV energies with modest laser powers. This material is based upon work supported by the Department of Energy Office of Fusion Energy under Award Numbers DE-SC0016253, DE-SC0019135, and DE-SC0017950.