The Maslov index is a powerful and well known tool in the study of Hamiltonian systems, providing a generalization of Sturm-Liouville theory to systems of equations. For non-Hamiltonian systems, one no longer has the symplectic structure needed to define the Maslov index. In this talk I will describe a recent construction of a "generalized Maslov index" for a very broad class of differential equations. The key observation is that the manifold of Lagrangian planes can be enlarged considerably without altering its topological structure, and in particular its fundamental group. This is joint work with Paul Cornwell, Chris Jones and Robert Marangell.