  The asymptotics of the probability that the self-intersection local time of a
random walk on $\Z^d$ exceeds its expectation by a large amount is a
fascinating subject because of its relation to some models from Statistical
Mechanics, to large-deviation theory and variational analysis and because of
the variety of the effects that can be observed. However, the proof of the
upper bound is notoriously difficult and requires various sophisticated
techniques. We survey some heuristics and some recently elaborated techniques
and results. This is an extended summary of a talk held on the CIRM-conference
on {\it Excess self-intersection local times, and related topics} in Luminy,
6-10 Dec., 2010.
