  We theoretically investigate a photonic Kagome lattice which can be realized
in microwave cavity arrays using current technology. The Kagome lattice
exhibits an exotic band structure with three bands one of which can be made
completely flat. The presence of artificial gauge fields allows to emulate
topological phases and induce chiral edge modes which can coexist inside the
energy gap with the flat band that is topologically trivial. By tuning the
artificial fluxes or in the presence of disorder, the flat band can also
acquire a bandwidth in energy allowing the coexistence between chiral edge
modes and bulk extended states; in this case the chiral modes become fragile
towards scattering into the bulk. The photonic system then exhibits equivalents
of both a quantum Hall effect without Landau levels, and an anomalous Hall
effect characterized by a non-quantized Chern number. We discuss experimental
observables such as local density of states and edge currents. We show how
synthetic uniform magnetic fields can be engineered, which allows an
experimental probe of Landau levels in the photonic Kagome lattice. We then
draw on semiclassical Boltzmann equations for transport to devise a method to
measure Berry's phases around loops in the Brillouin zone. The method is based
solely on wavepacket interference and can be used to determine band Chern
numbers or the photonic equivalent of the anomalous Hall response. We
demonstrate the robustness of these measurements towards on-site and
gauge-field disorder. We also show the stability of the anomalous quantum Hall
phase for nonlinear cavities and for (artificial) atom-photon interactions.
