  We study ``frustrated'' hopping models, in which at least one energy band, at
the maximum or minimum of the spectrum, is dispersionless. The states of the
flat band(s) can be represented in a basis which is fully localized, having
support on a vanishing fraction of the system in the thermodynamic limit. In
the majority of examples, a dispersive band touches the flat band(s) at a
number of discrete points in momentum space. We demonstrate that this band
touching is related to states which exhibit non-trivial topology in real space.
Specifically, these states have support on one-dimensional loops which wind
around the entire system (with periodic boundary conditions). A counting
argument is given that determines, in each case, whether there is band touching
or not, in precise correspondence to the result of straightforward
diagonalization. When they are present, the topological structure protects the
band touchings in the sense that they can only be removed by perturbations
which {\sl also} split the degeneracy of the flat band.
