The interaction between a growing and collapsing bubble and two types of compliant structures is explored numerically. The fluid motion is modelled as a potential flow and computed using a boundary element method. The structure is modelled with linear elastic theory and its response is computed using a finite element method. The fluid and structural models are coupled by matching the pressure and the normal component of the velocity at the fluid-structure interface using non-linear boundary conditions. The first type of compliant structure is a rigid wall covered by a composite coating. These calculations simulate the experiments of Shima et al. [1]. It is found that when the mass per unit area of the coating is large, the bubble moves toward the coating, while when the mass per unit area is small, the bubble moves away from the coating. The numerical results on bubble collapse height (the height above the wall at which the north and south poles of the bubble meet at the end of the calculation) are in good agreement with the experimental results and represent a significant improvement over past attempts to simulate this phenomenon. The distribution of mass through the thickness of the coating is found to be an important factor in the fluid-structure interaction. The second compliant structure is a neutrally buoyant, submerged, spherical shell. The effect of the initial distance between the shell and the bubble and the shell diameter and stiffness on the fluid-structure interaction is explored. It is found that for small initial distances between the bubble and the shell, the bubble moves toward the shell, while the shell moves away from the bubble as the bubble grows and toward the bubble as the bubble collapses. Both the rigid body motion and the deformation of the shell are larger for the smaller shells. For the range of geometries and shell properties considered, it is found that the bubble motion is affected more by the rigid body motion of the shell than the deformation of the shell.