  Euler systems are certain compatible families of cohomology classes, which
play a key role in studying the arithmetic of Galois representations. We
briefly survey the known Euler systems, and recall a standard conjecture of
Perrin-Riou predicting what kind of Euler system one should expect for a
general Galois representation. Surprisingly, several recent constructions of
Euler systems do not seem to fit the predictions of this conjecture, and we
formulate a more general conjecture which explains these extra objects. The
novel aspect of our conjecture is that it predicts that there should often be
Euler systems of several different ranks associated to a given Galois
representation, and we describe how we expect these objects to be related.
