We investigate prompt convection in core collapse supernovae and its consequences for the late-time shock evolution and supernova outcome. We examine the evolution of the core prior to the onset of convection and find that the negative entropy gradients imprinted on the outer core by the weakening shock, which, along with the negative lepton gradient, drive the convection, are very sensitive to (1) the neutrino transport and microphysics included in the core collapse simulation and (2) the nuclear equation of state. We perform a number of detailed one-dimensional spherically symmetric simulations of prompt convection using a mixing-length algorithm in a code coupling the core hydrodynamics with multigroup flux-limited diffusion of neutrinos of all types. We find that prompt convection does not have a significant effect on the neutrino luminosities or spectra in the postshock region and, consequently, on the late-time postshock neutrino heating and shock evolution. Consequently, we do not find that prompt convection is important for the supernova explosion mechanism.