We will give an overview of a set of methods being developed for solving classical partial differential equations (PDEs) in irregular geometries, or in the presence of free boundaries. In this approach, the irregular geometry is represented on a rectangular grid by specifying the intersection of each grid cell with the region on one or the other side of the boundary. This leads to a natural conservative discretization of the solution to the PDE on either side of the boundary. When the boundary is a solid wall for a fluid, these methods are often referred to as embedded boundary methods; more generally, they are called volume-of-fluid methods. Some of the recent developments in this area that we will discuss include: methods for elliptic free boundary value problems; automatic grid generation from implicit function representations of boundaries; high-order accurate methods; solving PDEs on surfaces; and software infrastructure issues involved in obtaining high performance while supporting a high degree of programming flexibility. We will also discuss applications examples from several fields of science and engineering.