  We describe a general strategy, PERM (Pruned-Enriched Rosenbluth Method), for
sampling configurations from a given Gibbs-Boltzmann distribution. The method
is not based on the Metropolis concept of establishing a Markov process whose
stationary state is the wanted distribution. Instead, it starts off building
instances according to a biased distribution, but corrects for this by cloning
"good" and killing "bad" configurations. In doing so, it uses the fact that
nontrivial problems in statistical physics are high dimensional. Therefore,
instances are built step by step, and the final "success" of an instance can be
guessed at an early stage. Using weighted samples, this is done so that the
final distribution is strictly unbiased. In contrast to evolutionary
algorithms, the cloning/killing is done without simultaneously keeping a large
population in computer memory. We apply this in large scale simulations of
homopolymers near the theta and unmixing critical points. In addition we sketch
other applications, notably to polymers in confined geometries and to randomly
branched polymers. For theta polymers we confirm the very strong logarithmic
corrections found in previous work. For critical unmixing we essentially
confirm the Flory-Huggins mean field theory and the logarithmic corrections to
it computed by Duplantier. We suggest that the latter are responsible for some
apparent violations of mean field behavior. This concerns in particular the
exponent for the chain length dependence of the critical density which is 1/2
in Flory-Huggins theory, but is claimed to be $\approx 0.38$ in several
experiments.
