Single compact scatterers insonified at frequencies near their intrinsic radial resonance ω0act primarily as monopole sources, amplifying the local pressure field by a factor of order (kRa)−1,abeing the scatterer radius andkRthe wavenumber at ω0in the surrounding material (kRa≂1.37×10−2for air bubbles in water). This paper deals withsystemsof such scatterers (bubbles, inflated balloons, thin shells in water) which, through couplingviamultiple scatter, may themselves exhibit true resonant modes at frequenciesclosetoω0, amplifying the local pressure field by a further factor of order (kRa)−1, leading to net amplifications in excess of 103for bubbles or balloons, i.e., of over 120 dB for the intensity. This phenomenon, referred to assuperresonance, is predicted by formulas for the equivalent source strengthBof each scatterer of the system, proper account being taken of multiple scatter between all elements, using a self‐consistent scheme. It has long been known that for a pair of scatterers (doublet), or in periodic lattices,Bexhibits maxima for certain values ofx=kl(lbeing the distance between scatterers), which had been called resonances [Twersky, J. Opt. Soc. Am.52, 145–171 (1962)], but are in fact only partial (quasi) resonances. It is demonstrated here that, under suitable conditions, true resonances exist; i.e., in the absence of attenuation,Bexhibitsrealpoles(the infinities are removed by introduction of acoustic damping and finite radii). The existence of such superresonances is furthered by the presence of elastic boundaries in the fluid near the system (e.g., a thin plate or a solid half‐space). Multiple scatter coupling then occurs by means of guided flexural or surface waves, which offer a more efficient energy transport mechanism between scatterers than the acoustic modes (volume waves) of a homogeneous fluid fullspace.In the latter case, a doublet of resonant bubbles, or shells, exhibits no superresonances (poles), only partial ones (quasiresonances); it takes at least three scatterers to create a true superresonant system in a fullspace, and even so there is only one resonant spacing (kl=1.8955, ω=1.0022 ω0for an equilateral triangle of bubbles in water). But near a thin plate, both doublet and triplet configurations develop spectra of resonant spacingsxnand frequencies ωn. Near an elastic solid half‐space, Stoneley (Scholte) wave coupling creates similar effects. The role of small interface roughness on this phenomenon is also examined.