  All known realizations of optical wave packets that accelerate along their
propagation axis, such as Airy wave packets in dispersive media or
wave-front-modulated X-waves, exhibit a constant acceleration; that is, the
group velocity varies linearly with propagation. Here we synthesize space-time
wave packets that travel in free space with arbitrary axial acceleration
profiles, including group velocities that change with integer or fractional
exponents of the distance. Furthermore, we realize a composite acceleration
profile: the wave packet first accelerates from an initial to a terminal group
velocity, decelerates back to the initial value, and then travels at a fixed
group velocity. These never-before-seen optical-acceleration phenomena are all
produced using the same experimental arrangement that precisely sculpts the
wave packet's spatio-temporal spectral structure.
