Suppose we have n players who wish to jointly perform a quantum computation, but some of their number are untrustworthy and are trying to learn privileged information and/or sabotage the computation. How many cheaters can we tolerate and still have a secure protocol? Secure multiparty classical computation is possible with fewer than n/3 cheaters, or fewer than n/2 when a secure broadcast channel is available. It turns out the same bounds hold in the quantum case. I will sketch the construction, which owes a great deal to the classical solutions, but requires a number of new components as well. This talk describes joint work with Ben-Or, Crepeau, Hassidim, and Smith.