We study the capacity of rewritable storage cells with discrete memoryless write channels. We consider a constraint on the maximum number of rewrites and give a closed-form expression for the capacity of cells with symmetric write channels. For binary symmetric write channels this result is extended to the case where only noisy observations of the state are available to the encoder and decoder. We also consider a constraint on the expected number of rewrites and derive a lower bound on the capacity. This bound coincides with the capacity for cells with binary symmetric write channels.