  The dynamics of two-dimensional thin premixed flames is addressed in the
framework of mathematical models where the flow field on either side of the
front is piecewise incompressible and vorticity-free. Flames confined in
channels with asymptotically-straight impenetrable walls are considered. Beside
a few free propagations along straight channels, attention is focused on flames
propagating against high-speed flows and positioned near a round central
obstacle, or near two symmetric bumps protruding inward. Combining conformal
maps and Green's functions, a regularised generalisation of Frankel's
integro-differential equation for the instantaneous front shape in each
configuration is derived, and solved numerically. This produces a variety of
real looking phenomena: steady fronts (symmetric or not), noise-induced
sub-wrinkles, flash-back events and breathing fronts in pulsating flows.
Perspectives and open mathematical/physical problems are finally evoked.
