  We consider the cubic-quintic nonlinear Schr{\"o}dinger equation in space
dimension up to three. The cubic nonlinearity is thereby focusing while the
quintic one is defocusing, ensuring global well-posedness of the Cauchy problem
in the energy space. The main goal of this paper is to investigate the
interplay between dispersion and orbital (in-)stability of solitary waves. In
space dimension one, it is already known that all solitons are orbitally
stable. In dimension two, we show that if the initial data belong to the
conformal space, and have at most the mass of the ground state of the cubic
two-dimensional Schr{\"o}dinger equation, then the solution is asymptotically
linear. For larger mass, solitary wave solutions exist, and we review several
results on their stability. Finally, in dimension three, relying on previous
results from other authors, we show that solitons may or may not be orbitally
stable.
