  We review recent progress on Horn's problem, which asks for a description of
the possible eigenspectra of the sum of two matrices with known eigenvalues.
  After revisiting the classical case, we consider several generalizations in
which the space of matrices under study carries an action of a compact Lie
group, and the goal is to describe an associated probability measure on the
space of orbits. We review some recent results about the problem of computing
the probability density via orbital integrals and about the locus of
singularities of the density. We discuss some relations with representation
theory, combinatorics, pictographs and symmetric polynomials, and we also
include some novel remarks in connection with Schur's problem.
