The immersed boundary (IB) method is a mathematical formulation and numerical scheme for problems of fluid-structure interaction. The IB method blurs the distinction between solid and fluid, so that both can be treated in a unified way. The mathematical formulation involves both the Lagrangian description of the motion that is fundamental to elasticity theory and also the Eulerian description that is commonly used in fluid mechanics. These are linked by interaction equations involving the Dirac delta function. The numerical scheme derived from this formulation employs a Cartesian grid for Eulerian variables and a moving curvilinear mesh for Lagrangian variables. Data transfer between these two meshes involves a regularized Dirac delta function, with properties chosen to preserve key physical quantities such as momentum and angular momentum. Results from example applications will be shown as computer animations.