  This is a review paper outlining recent progress in the spectral analysis of
first order systems. We work on a closed manifold and study an elliptic
self-adjoint first order system of linear partial differential equations. The
aim is to examine the spectrum and derive asymptotic formulae for the two
counting functions. Here the two counting functions are those for the positive
and the negative eigenvalues. One has to deal with positive and negative
eigenvalues separately because the spectrum is, generically, asymmetric.
