  We present a comprehensive review about the various facets of kink solutions
with a power law tail which have received considerable attention during the
last few years. This area of research is in its early stages and while several
aspects have become clear by now, there are a number of issues which have only
been partially understood or not understood at all. We first discuss the
aspects which are reasonably well known and then address in some detail the
issues which are only partially or not understood at all. We present a wide
class of higher (than sixth) order field theory models admitting implicit kink
as well as mirror kink solutions where the two tails facing each other have a
power law or a power-tower type fall off while the other two ends not facing
each other could have either an exponential or a power law tail. The models
admitting implicit kink solutions where the two ends facing each other have an
exponential tail while the other two ends have a power law tail are also
discussed. Moreover, we present several field theory models which admit
explicit kink solutions with a power law fall off. We note that in all the
polynomial models while the potential $V(\phi)$ is continuous, its derivative
is discontinuous. We also discuss one of the most important and only partially
understood issues of the kink-kink and the kink-antikink forces in case the
tails facing each other have a power law fall off. Finally, we briefly discuss
the kink-antikink collisions at finite velocity and present some open
questions.

Pattern Formation and Solitons;High Energy Physics - Phenomenology