  The domination polynomial (the total domination polynomial) of a graph $ G $
of order $ n $ is the generating function of the number of dominating sets
(total dominating sets) of $ G $ of any size. In this paper, we study the
domination polynomial and the total domination polynomial of zero-divisor
graphs of the ring $ \mathbb{Z}_n $ where $ n\in\lbrace 2p, p^2, pq, p^2q, pqr,
p^\alpha\rbrace $, and $ p, q, r $ are primes with $ p>q>r>2 $.
