  We present a general framework to tackle quantum optics problems with giant
atoms, i.e. quantum emitters each coupled {\it non-locally} to a structured
photonic bath (typically a lattice) of any dimension. The theory encompasses
the calculation and general properties of Green's functions, atom-photon bound
states (BSs), collective master equations and decoherence-free Hamiltonians
(DFHs), and is underpinned by a formalism where a giant atom is formally viewed
as a normal atom lying at a fictitious location. As a major application, we
provide for the first time a general criterion to predict/engineer DFHs of
giant atoms, which can be applied both in and out of the photonic continuum and
regardless of the structure or dimensionality of the photonic bath. This is
used to show novel DFHs in 2D baths such as a square lattice and photonic
graphene.
