We describe a very general abstract form of sieve based on a large sieve
inequality which generalizes both the classical sieve inequality of Montgomery
(and its higher-dimensional variants), and our recent sieve for Frobenius over
function fields. The general framework suggests new applications. We get some
first results on the number of prime divisors of ``most'' elements of an
elliptic divisibility sequence, and we develop in some detail ``probabilistic''
sieves for random walks on arithmetic groups, e.g., estimating the probability
of finding a reducible characteristic polynomial at some step of a random walk
on SL(n,Z). In addition to the sieve principle, the applications depend on
bounds for a large sieve constant. To prove such bounds involves a variety of
deep results, including Property (T) or expanding properties of Cayley graphs,
and the Riemann Hypothesis over finite fields. It seems likely that this sieve
can have further applications.
