Mesh generation for traditional finite element analysis has proven to be very difficult to fully automate especially using hexahedral elements for complex 3D geometry. Several modifications to the finite element method (FEM), such as the meshless methods, have been proposed for avoiding mesh generation. An alternative approach has recently gained popularity where the geometry, created as a solid model in CAD software, is embedded or immersed in a non-conforming background mesh for analysis. In this approach, referred to here as the immersed boundary approach, a background mesh that is independent of the geometry is used for piece-wise interpolation or approximation of the solution. Therefore, a uniform mesh with regular shaped or undistorted elements can be used and such a mesh is easy to generate automatically. When the geometry is immersed in the background mesh, the boundary elements are often only partly inside the geometry and the nodes of the mesh may not be on the boundaries. Many new methods have been developed to integrate over partial elements and to apply boundary and interface conditions when the boundaries of the geometries do not conform to the background mesh. These methods are reviewed in this article with particular emphasis on the implicit boundary method and step boundary method for applying boundary conditions. In addition, B-spline elements and several applications of the immersed boundary approach are surveyed including composite microstructures and structural elements for plates and shells.