  We present a method for constructing families of isospectral systems, using
linear representations of finite groups. We focus on quantum graphs, for which
we give a complete treatment. However, the method presented can be applied to
other systems such as manifolds and two-dimensional drums. This is demonstrated
by reproducing some known isospectral drums, and new examples are obtained as
well. In particular, Sunada's method is a special case of the one presented.
