  Certain tight binding lattices host macroscopically degenerate flat spectral
bands. Their origin is rooted in local symmetries of the lattice, with
destructive interference leading to the existence of compact localized
eigenstates. We study the robustness of this localization to disorder in
different classes of flat band lattices in one and two dimensions. Depending on
the flat band class, the flat band states can either be robust, preserving
their strong localization for weak disorder W, or they are destroyed and
acquire large localization lengths $\xi$ that diverge with a variety of
unconventional exponents $\nu$, $\xi \sim 1/W^{\nu}$.
