By using some fundamental knowledge,research methods and research results about the theories of value distribution and normal family for meromorphic functions and based on Gu's theorem,a criterion of normality concerning meromorphic functions whose higher zeros are not fixed functions was provided,and is described as follows:let k≥2 be an integer and F be a family of meromorphic functions on a unit disc Δ,all the zeros of every f which belongs to F have multiplicity at least k and all the poles have multiplicity at least 3,h(z) be a holomorphic functions,and h(z)≠0,and assume that the following two conditions that f(z)=0⇒|f^(k)|≤A,A is a positive number,and f^(k)(z)≠h(z) hold for every f∈F,then F is normal in Δ.