  We consider an Allen-Cahn type equation with a bistable nonlinearity
associated to a double-well potential whose well-depths can be slightly
unbalanced, and where the coefficient of the nonlinear reaction term is very
small. Given rather general initial data, we perform a rigorous analysis of
both the generation and the motion of interface. More precisely we show that
the solution develops a steep transition layer within a small time, and we
present an optimal estimate for its width. We then consider a class of
reaction-diffusion systems which includes the FitzHugh-Nagumo system as a
special case. Given rather general initial data, we show that the first
component of the solution vector develops a steep transition layer and that all
the results mentioned above remain true for this component.
