The complete homogeneous symmetric (CHS) polynomials can be used to define a family of norms on Hermitian matrices. These ‘CHS norms’ are peculiar in the sense that they depend only on the eigenvalues of a matrix and not its singular values (as opposed to the Ky-Fan and Schatten norms). We will first give a general overview behind the construction of these norms (as well as their extensions to all n x n complex matrices). The construction and validation of these norms will take us on a tour of probability theory, convexity analysis, partition combinatorics and trace polynomials in noncommuting variables. We then discuss open problems and potential for future work. This talk is based on joint work with Konrad Aguilar, Stephan Garcia and Jurij Volčič.