  It has been shown previously that an extremal Reissner-Nordstr\"om or an
extremal Kerr black hole cannot be overcharged or overspun by a test particle
if radiative and self-force effects are neglected. In this paper, we consider
extremal charged and rotating black holes with cosmological constants. By
studying the motion of test particles, we find the following results: An
extremal Reissner-Nordstr\"om anti-de Sitter (RN-AdS) black hole can be
overcharged by a test particle but an extremal Reissner-Nordstr\"om de Sitter
(RN-dS) black hole cannot be overcharged. We also show that both extrmal
Kerr-de-Sitter (Kerr-dS) and Kerr-anti-de-Sitter (Kerr-AdS) black holes can be
overspun by a test particle, implying a possible breakdown of the cosmic
censorship conjecture. For the Kerr-AdS case, the overspinning requires that
the energy of the particle be negative, a reminiscent of the Penrose process.
In contrast to the extremal RN and Kerr black holes, in which cases the cosmic
censorship is upheld, our results suggest some subtle relations between the
cosmological constants and the comic censorship. We also discuss the effect of
radiation reaction for the Kerr-dS case and find that the magnitude of energy
loss due to gravitational radiation may not be enough to prevent the violation
of the cosmic censorship.
