  These are lecture notes of a course given at the 9th International Summer
School on Fundamental Problems in Statistical Mechanics, held in Altenberg,
Germany, in August 1997. In these notes, we discuss at an elementary level
three themes concerning interface dynamics that play a role in pattern forming
systems: (i) We briefly review three examples of systems in which the normal
growth velocity is proportional to the gradient of a bulk field which itself
obeys a Laplace or diffusion type of equation (solidification, viscous fingers
and streamers), and then discuss why the Mullins-Sekerka instability is common
to all such gradient systems. (ii) Secondly, we discuss how underlying an
effective interface description of systems with smooth fronts or transition
zones, is the assumption that the relaxation time of the appropriate order
parameter field(s) in the front region is much smaller than the time scale of
the evolution of interfacial patterns. Using standard arguments we illustrate
that this is generally so for fronts that separate two (meta)stable phases: in
such cases, the relaxation is typically exponential, and the relaxation time in
the usual models goes to zero in the limit in which the front width vanishes.
(iii) We finally summarize recent results that show that so-called ``pulled''
or ``linear marginal stability'' fronts which propagate into unstable states
have a very slow universal power law relaxation. This slow relaxation makes the
usual ``moving boundary'' or ``effective interface'' approximation for problems
with thin fronts, like streamers, impossible.
