A k-uniform hypergraph G = ( V;E ) is a set of vertices V ? N together with a family E of k-element subsets of V, which are called edges. In this note, by v ( G ) = | V | and e ( G ) = | E | we denote the number of vertices and edges of G=(V, E), respectively. By a matching we mean any family of disjoint edges of G, and we denote by µ ( G ) the size of the largest matching contained in E. Moreover, by v _ k ( n; s ) we mean the largest possible number of edges in a k-uniform hypergraph G with v ( G ) = n and µ ( G ) = s , and by M_ k ( n; s ) we denote the family of the extremal hypergraphs for this problem, i.e. H I M k ( n; s ) if v ( H ) = n , µ ( H ) = s , and e ( H ) = n _ k ( n; s ). In 1965 Erd?s conjectured that, unless n = 2k and s = 1, all graphs from M_ k ( n; s ) are either cliques, or belong to the family Cov_ k ( n; s ) of hypergraphs on n vertices in which the set of edges consists of all k-subsets which intersect a given subset S ? V , with |S| = s. This conjecture, which is a natural generalization of Erd?s-Gallai result for graphs, has been verified only for k = 3. For general k there have been series of results which state that M_ k ( n; s ) = Cov_ k ( n; s ) for g ( k ) s where g(k) is some function of k. The existence of such g(k) was shown by Erd?s, then ollobas, Daykin and Erd?s proved that this equation holds whenever g ( k ) 2 k^ 3; Frankl and Füredi showed that the equation is true for g ( k ) 100 k^ 2 and recently, Huang, Loh, and Sudakov verified its truness for g ( k ) 3 k^ 2. Also, Peter Frankl slightly improved these bounds and confirmed that for g ( k ) 2 k^ 2/ log ( k ) , M k ( n; s ) = Cov k ( n; s ). The problem of finding the maximum matching in a hypergraph has many applications in various different areas of mathematics, computer science, and even computational chemistry. Yet although the graph matching problem is fairly well understood, and solvable in polynomial time, most of the problems related to hypergraph matching tend to be very difficult and remain unsolved. Indeed, the hypergraph matching problem is known to be NP-hard even for 3-uniform hypergraphs, without any good approximation algorithm. In this dissertation, we would like to explore the process of tries to prove the Erd?s matching conjecture. In this regard, we will give the proof of Sudakov which showed that g ( k ) 3 k^ 2 in Chapter 2. After that, in Chapter 3, the conjecture is proved for 3-uniform hypergraphs. Finally, we investigate the last known result on this conjecture which says that for g ( k ) 2 k^ 2/ log ( k ) , we have M_k ( n; s ) = Cov_k ( n; s ).