More than a decade after the discovery of deep episodic slow slip and tremor, or slow earthquakes, at subduction zones, much research has been carried out to investigate the structural and seismic properties of the environment in which they occur. Slow earthquakes generally occur on the megathrust fault some distance downdip of the great earthquake seismogenic zone in the vicinity of the mantle wedge corner, where three major structural elements are in contact: the subducting oceanic crust, the overriding forearc crust and the continental mantle. In this region, thermo-petrological models predict significant fluid production from the dehydrating oceanic crust and mantle due to prograde metamorphic reactions, and their consumption by hydrating the mantle wedge. These fluids are expected to affect the dynamic stability of the megathrust fault and enable slow slip by increasing pore-fluid pressure and/or reducing friction in fault gouges. Resolving the fine-scale structure of the deep megathrust fault and the in situ distribution of fluids where slow earthquakes occur is challenging, and most advances have been made using teleseismic scattering techniques (e.g., receiver functions). In this paper we review the teleseismic structure of six well-studied subduction zones (three hot, i.e., Cascadia, southwest Japan, central Mexico, and three cool, i.e., Costa Rica, Alaska, and Hikurangi) that exhibit slow earthquake processes and discuss the evidence of structural and geological controls on the slow earthquake behavior. We conclude that changes in the mechanical properties of geological materials downdip of the seismogenic zone play a dominant role in controlling slow earthquake behavior, and that near-lithostatic pore-fluid pressures near the megathrust fault may be a necessary but insufficient condition for their occurrence.