Review of topological recursion, quasi-polynomiality and rationality results for various types of Hurwitz numbers

We review results on spectral curve, topological recursion, quasi-polynomiality and rationality for various types of Hurwitz numbers, including simple, monotone, and r-spin Hurwitz numbers, Bousquet-Mélou–Schaeffer numbers, coefficients of Ooguri-Vafa partition functions for colored HOMFLY polynomials of knots, etc.