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For a monoid M, we introduce M-McCoy rings, which are generalization of McCoy rings, and we investigate their properties. Every M-Armendariz ring is M-McCoy for any monoid M. We show that R is an M-McCoy ring if and only if an n × n upper triangular matrix ring aUTn (R) over R is an M-McCoy ring for any monoid M. It is proved that if R is McCoy and R[x] is M-McCoy, then RIM] is McCoy for any monoid M. Moreover, we prove that if R is M-McCoy, then RIM] and R[x] are M-McCoy for a commutative and cancellative monoid M that contains an infinite cyclic submonoid.