  Exceptional point and spectral singularity are two types of singularity that
are unique to non-Hermitian systems. Here, we report the high-order spectral
singularity as a high-order pole of the scattering matrix for a non-Hermitian
scattering system, and the high-order spectral singularity is a unification of
the exceptional point and spectral singularity. At the high-order spectral
singularity, the scattering coefficients have high-order divergence and the
scattering system stimulates high-order lasing. The wave emission intensity is
polynomially enhanced, and the order of the growth in the polynomial intensity
linearly scales with the order of the spectral singularity. Furthermore, the
coherent input controls and alters the order of the spectral singularity. Our
findings provide profound insights into the fundamentals and applications of
high-order spectral singularities.
