  We survey some of the major results about normal Hilbert polynomials of
ideals. We discuss a formula due to Lipman for complete ideals in regular local
rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in
Cohen-Macaulay analytically unramified local rings. We also discuss recent
works of Goto-Hong-Mandal and Mandal-Singh-Verma concerning the positivity of
the first coefficient of the normal Hilbert polynomial in unmixed analytically
unramified local rings. Results of Moral{\'e}s and Villarreal linking normal
Hilbert polynomial of monomial ideals with Ehrhart polynomials of polytopes are
also presented.
