  The powerset construction is a standard method for converting a
nondeterministic automaton into a deterministic one recognizing the same
language. In this paper, we lift the powerset construction from automata to the
more general framework of coalgebras with structured state spaces. Coalgebra is
an abstract framework for the uniform study of different kinds of dynamical
systems. An endofunctor F determines both the type of systems (F-coalgebras)
and a notion of behavioural equivalence (~_F) amongst them. Many types of
transition systems and their equivalences can be captured by a functor F. For
example, for deterministic automata the derived equivalence is language
equivalence, while for non-deterministic automata it is ordinary bisimilarity.
We give several examples of applications of our generalized determinization
construction, including partial Mealy machines, (structured) Moore automata,
Rabin probabilistic automata, and, somewhat surprisingly, even pushdown
automata. To further witness the generality of the approach we show how to
characterize coalgebraically several equivalences which have been object of
interest in the concurrency community, such as failure or ready semantics.
