We consider planar, uniformly propagating combustion waves driven by the filtration of gas containing an oxidizer which reacts with the combustible porous medium through which it moves. We find that these waves are typically unstable with respect to hydrodynamic perturbations. For both forward (coflow) and reverse (counterflow) filtration combustion (FC), in which the direction of gas flow is the same as or opposite to the direction of propagation of the combustion wave, respectively, the basic mechanism leading to instability is the reduction of the resistance to flow in the region of the combustion products, due to an increase of the porosity in that region. Another destabilizing effect in forward FC is the production of gaseous products in the reaction. In reverse FC this effect is stabilizing. We also describe an alternative mode of propagation, in the form of a finger propagating with constant velocity. The finger region occupied by the combustion products is separated from the unburned region by a front, in which chemical reactions and heat and mass transport occur. We show that the finger solution of the combustion problem can be characterized as a solution of a Saffman-Taylor (ST) problem, originally formulated to describe the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele-Shaw configuration. The ST problem is known to possess a family of finger solutions, with each member characterized by its own velocity and each occupying a different fraction of the porous channel through which it propagates. We propose a criterion to select the correct member of the family of solutions, based on consideration of the ST problem itself, rather than on modifications of the problem, e.g., by adding surface tension to the model and then taking the limit of vanishing surface tension.