This paper is an up-to-date introduction to the problem of uniqueness versus
non-uniqueness of infinite clusters for percolation on Zd and,
more generally, on transitive graphs. For iid percolation on Zd,
uniqueness of the infinite cluster is a classical result, while on certain
other transitive graphs uniqueness may fail. Key properties of the graphs in
this context turn out to be amenability and nonamenability. The same problem is
considered for certain dependent percolation models -- most prominently the
Fortuin--Kasteleyn random-cluster model -- and in situations where the standard
connectivity notion is replaced by entanglement or rigidity. So-called
simultaneous uniqueness in couplings of percolation processes is also
considered. Some of the main results are proved in detail, while for others the
proofs are merely sketched, and for yet others they are omitted. Several open
problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org