In this article the radiation and scattering of sound from elastic bodies of arbitrary shape are examined. The coupled acoustic/elastic boundary value problem is reformulated into a pair of integral equations, one for the acoustic domain (the host fluid) and one for the elastic body. Both of these integral equations are valid at the interface (the body’s surface) between the elastic body and the fluid. The integral equations are coupled by continuity conditions at the interface. The integral equations are reduced to numerical form using quadratic isoparametric elements and solved simultaneously to determine the unknown pressure and displacements on the interface. Field solutions are obtained by quadrature of the data on the interface. The integral equation method is tested for several radiation and scattering problems in which the elastic body is either a hollow or solid sphere of various material compositions.