Let \(R[x]\) be a polynomial ring over a ring \(R\). Then \(R\) is called a right McCoy (respectively left McCoy) ring if \(f(x)g(x)=0\) for each \(f(x),g(x)\neq 0\in R[x]\) implies that \((f(x))r=0\) for some \(r\neq 0\in R\) (respectively \(rg(x)=0\)), a McCoy ring is both left and right McCoy. The authors show some standard ring theoretic properties and relations for some classes of rings. An example of a ring \(R\) is constructed such that \(R\) is a right McCoy, but the power series ring \(R[\![t]\!]\) is not. It is shown that the direct product of rings is (right) McCoy if and only if so is each factor. For fixed positive integers \(m\) and \(n\), a ring \(R\) is called \((m,n)\)-right McCoy if for any nonzero \(f(x),g(x)\in R[x]\) with \(\deg f(x)=m\), \(\deg g(x)=n\), \(f(x)g(x)=0\) implies that \((f(x))r=0\) for some \(r\neq 0\in R\). Then semicommutative rings (i.e., \(ab=0\) implies \(aRb=0\) for all \(a,b\in R\)) are \((1,1)\)- and \((2,2)\)-McCoy, but not fully McCoy. Also, there exist McCoy rings with 1 which are not Abelian (i.e., all idempotents are central). A ring is called right duo if all right ideals are two-sided. Then right duo rings are right McCoy. A ring is called 2-primal if every nilpotent element is contained in the lower nilradical. Then a 2-primal ring \(R\) is either right McCoy, or \(f(x)g(x)=0\) with nonzero \(f(x),g(x)\in R[x]\) implies that \((g(x))r=0\) for some \(r\neq 0\in R\). Moreover, it is shown that matrix rings and upper triangular rings over a nonzero ring are not \((1,1)\)-McCoy.