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In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and \({\mathcal{F}}\) a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair \({f(z),g(z)\in \mathcal{F}}\) and one of the following conditions holds: (1) P(z) − α(z0) has at least three distinct zeros for any \({z_{0}\in D}\); (2) There exists \({z_{0}\in D}\) such that P(z) − α(z0) has at most two distinct zeros and α(z) is nonconstant. Assume that β0 is a zero of P(z) − α(z0) with multiplicity p and that the multiplicities l and k of zeros of f(z) − β0 and α(z) − α(z0) at z0, respectively, satisfy k ≠ lp, for all \({f(z)\in\mathcal{F}}\). Then \({\mathcal{F}}\) is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.