The performance of a secure communication system such as the wiretap channel is usually characterized by its secrecy capacity. In this paper, the issue of whether or not the secrecy capacity is a continuous function of the system parameters is examined. In particular, this is done for channel uncertainty modeled via compound channels and arbitrarily varying channels, in which the legitimate users know only that the true channel realization is from a pre-specified uncertainty set. In the former model, this realization remains constant for the entire duration of transmission, while in the latter the realization varies from channel use to channel use in an unknown and arbitrary manner. The secrecy capacity of the compound wiretap channel is shown to be robust in the sense that it is a continuous function of the uncertainty set. Thus, small variations in the uncertainty set lead to small variations in secrecy capacity. However, the secrecy capacity of the arbitrarily varying wiretap channel is shown to be discontinuous in the uncertainty set meaning that small variations can lead to dramatic losses in capacity.