  We present theory and experiments on the dynamics of reaction fronts in
two-dimensional, vortex-dominated flows, for both time-independent and
periodically driven cases. We find that the front propagation process is
controlled by one-sided barriers that are either fixed in the laboratory frame
(time-independent flows) or oscillate periodically (periodically driven flows).
We call these barriers \emph{burning invariant manifolds} (BIMs), since their
role in front propagation is analogous to that of invariant manifolds in the
transport and mixing of passive impurities under advection. Theoretically, the
BIMs emerge from a dynamical systems approach when the
advection-reaction-diffusion dynamics is recast as an ODE for front element
dynamics. Experimentally, we measure the location of BIMs for several
laboratory flows and confirm their role as barriers to front propagation.
