We present a new method to obtain high-mobility three-dimensional electron gas systems. We have achieved control of carrier density and of carrier profile by growth of the first remotely-doped parabolic potential well structures. Computer-controlled molecular beam epitaxy is used to grow a layer of ultra-fine superlattices with a programmable composition gradient. This produces conduction-band potentials which, in the absence of doping, are equivalent to the potential profiles of fixed charge distributions. When conduction electrons are introduced into these graded wells through remote doping of the barrier regions, they distribute themselves in such a way as to produce a uniform chemical potential at thermal equilibrium. We illustrate through computer simulations employing Fermi statistics that electrons introduced into a wide parabolic potential well distribute themselves uniformly. More significantly, the carrier distribution in the well is remarkably insensitive to the dopant sheet charge in the barrier, the more so at lower temperatures. We have fabricated remotely-doped graded potential well structures of the proposed type by molecular beam epitaxy. These structures exhibit the above effects. Measured mobilities of such threedimensional electron gases grown using the GaAs/AlxGal_xAs system are higher than those of bulk-doped GaAs doped to give the same uniform electron concentration.