Gibbard's theorem deals with processes of collective choice that may not be ordinal, i.e. where a voter's action may not consist in communicating a preference order over the candidates.
Gibbard's 1978 theorem and
Hylland's theorem extend these results to non-deterministic mechanisms, i.e. where the outcome may not only depend on the ballots but may also involve a part of chance.
The Duggan–Schwartz theorem extend this result in another direction, by dealing with deterministic voting rules that choose a nonempty subset of the candidates rather than a single winner.