Fusion categories appear in many areas of mathematics. They are realized by topological quantum field theories, representations of finite groups and Hopf algebras, and invariants for knots and Murray-von Neumann subfactors. An important numerical invariant of these categories are the Frobenius-Schur indicators, which are generalized versions of those for finite group representations. It is thought that these indicators should provide a complete invariant for a fairly wide class of fusion categories; in this talk we will discuss new families of so-called near-group fusion categories (i.e. those with only one non-invertible indecomposable object) which satisfy this property.