We investigate adaptive finite element methods for low Mach, steady, laminar combustion. The finite element discretization of the flame equations involves least squares control of streamline derivatives and pressure-velocity coupling as well as a new shock capturing term based on nonlinear crosswind diffusion yielding a suitable discrete maximum principle for the discrete solution. A posteriori error estimates derived from the dual weighted residual method are used to refine the mesh adaptively. Numerical results are presented for a Bunsen flame with simple chemistry on locally refined as well as fully unstructured Delaunay meshes. Solution quality is evaluated in terms of overall flame characteristics - including length, lift off and width - as well as undershoots in species and temperature profiles