  The theory of linear dispersive equations predicts that waves should spread
out and disperse over time. However, it is a remarkable phenomenon, observed
both in theory and practice, that once nonlinear effects are taken into
account, \emph{solitary wave} or \emph{soliton} solutions can be created, which
can be stable enough to persist indefinitely. The construction of such
solutions is relatively straightforward, but the fact that they are
\emph{stable} requires some significant amounts of analysis to establish, in
part due to symmetries in the equation (such as translation invariance) which
create degeneracy in the stability analysis. The theory is particularly
difficult in the \emph{critical} case in which the nonlinearity is at exactly
the right power to potentially allow for a self-similar blowup. In this article
we survey some of the highlights of this theory, from the more classical
orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the
more recent asymptotic stability and blowup analysis of Martel-Merle and
Merle-Raphael, as well as current developments in using this theory to
rigorously demonstrate controlled blowup for several key equations.
