| In 1920s, R. Nevanlinna introduced the characteristic functions of meromorphic functions and gave the famous Nevanlinna theory which is one of the greatest achivemeuts in mathematics in the 20th century. This theory is considered to be the basis of the morden meromorphic function theory, and it has a very important effect on the development and syncretic of many mathematical branches.The theorem of normal family has an ample development, which can be divided into three stages. It has not only academic meanings but also important application values to study the normal family. The third stage must track back to 1975 when L. Zaleman published a small paper. He built up an additional way to get the sufficient and neccessary condition of the not normal family from the normal criteria. The method makes a new wold in normal family, which is called Zalcman-Pang Method. It make the former method simple, and build up a lot of new normality criteria.It is a significant subject to consider the combination of normal family and shared values or functions. The work in this held wa.s first studied by Schwick. After that, many home or oversea scholars have a further study.This paper will introdece the main results on the subject in recently years, which consists of three parts.In chapter 1. we will brieftly introduce some fundamental results, definitions and some notations. In chapter 2, we mainly introduce some normality criterias concerning shared two values and a three-element set which have been obtained seperately by Pang xuecheng, L. Zalcman. and Liu Xiaojun, Pang xuecheng, some by L. Zalcman. Fang Mingliang, some normal family of the holomorphic functions sharing the fix points between its first derivative and the second derivative by Chang Jianming, Fang Mingliang, and the normality criteria concerning sharing values with their derivatives by Zhang Qingcai.In chapter 3, we study the normality criteria from the point of view of shared set of values.The main results are in the following.Theorem 1. Let F be a famiy of functions meromorphic on D in C, S = {a1, a2, a3} be a set which consists of three different pairwise finite complex numbers. Let k be a positive integer and d be a finite complex constant. Suppose that for each f∈F, f and f(k) share the set S and all the zeros of f - d have multiplicity at least k +1. Then F is a normal family on D.Corollary 1. Let F be a famiy of functions meromorphic on D in C. S = {a1,a2,a3} be a set which consists of three different pairwise finite complex numbers and k be a positive integer. Suppose that for each f∈F, f and f(k) share the set S and all the zeros of f - a3 have multiplicity at least k + 1. Then F is a normal family on D.Theorem 1 . Let F be a famiy of functions meromorphic on D in C. S = {a1, a2,a3} be a set which consists of three different pairwise finite complex numbers and d (?) S be a finite complex constant. Let k be a positive integer and M be a positeve number. Suppose that for each f∈F, f and f(k) share the set S, all the zeros of f - d have multiplicity at least k and |f(k)(z)|≤M whenever f(z) = d. Then F is a normal family on D.Theorem 2. Let F be a famiy of functions meromorphic on D in C, S -{a1, a2,a3} be a set which consists of three different pairwise finite complex numbers and k be a positive integer. Suppose that for each f∈F,f and f(k) share the set S, and for each ai∈S, the zeros of f - ai have multiplicity at least k. Then F is a normal family on D.Corollary 2. Let F be a famiy of functions holomorphic on D in C, S = {a1,a2} be a set which consists of two different finite complex constants and k > 1 be a positive integer. Suppose that for each f∈F, f and f(k) share the set S and for each ai∈S, the multiplicites of the zeros of f - a, are not less than k. Then F is a normal family on D.Theorem 3. Let F be a famiy of functions meromorphic on D in C, all of whose zeros have multiplicity of at least k, k be a positive integer, and 5 be a set which consists of at least k + 4 different pairwise finite complex numbers. Let M be a positeve number. Suppose that for each f∈F, |f(k)(z)|≤M whenever f(z) = ai, (ai∈6 5). Then F is a normal family on D.Theorem 4. Let F be a famiy of functions holomorphic on D in C and a1,a2 be two different finite complex constants. Let k > 1 be a positive integer. Suppose that for each f∈F, f and f(k) share the set S = {a1,a2} and the multiplicites of the zeros of f - a1 are not less than k. Then T is a normal family on D.Theorem 4 . Let F be a famiy of functions holomorphic on D in C, S -{a1,a2,a3} be a set which consists of three different pairwise finite complex numbers and k be a positive integer. Suppose that for each f∈F,f and f(k) share the set S and the multiplicites of the zeros of f - a1 are not less than k. Then F is a normal family on D.Theorem 5. Let f be a holomorphic function in complex plain. Suppose that f and f(k) share the set S = {1, -1} and f≠0. Then f = f(k) or f = -f(k).Theorem 6. Let f be a meromorphic function in complex plain, a be a nonzero complex constant, k be a positive integer. Suppose that f and f(k) share the set S = {a, -a} and 5δ(0, f) + (2k + 3)(?)(∞, f) > 2k + 7. Then f = f(k) or f = -f(k).
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