Let H_1, H_2,…, H_t be arbitrary k-uniform hypergraphs. The hypergraph Ramsey number R( H_1, H_2,…, H_t) is the smallest integer N such that in every t-coloring of the hyperedges Of the complete k-uniform hypergraph there is a monochromatic copy of H_i in color i, for some i, 1?i? t. In this thesis, we provide the exact value of the multicolor Ramsey number R(P_{n_1}, P_{n_2},…, P_{n_t},C_k) for certain values of n_i and k, where P_n and C_ denote a path and a cycle on n vertices, respectively. In addition, we determine the 2-color hypergraph Ramsey number of a k-uniform loose 3-cycle or 4-cycle: R(C^k_3,C^k_3)=3k-2 and R(C_4^k,C_4^k)=4k-3 (for k? 3). For more than three colors we could prove only that $R(C^3_3,C^3_3,C^3_3)=8. Nevertheless, the r-color Ramsey number of triangles for hypergraphs are much smaller than for graphs: for r? 3, r+5? R(C_3^3,C_3^3,…,C_3^3)? 3r.