The main goal of this thesis was to study the connection between the crystal basis theory of finite dimensional \(U_q(\mathfrak{gl}_n)\)-modules and combinatorics of Young diagrams and Young tableaux. In the first chapter we studied basic concepts and results from the field of Lie algebras and representations of related quantum groups. In the second chapter we introduced Kashiwara’s operators and the notion of crystal basis, which we used to state the tensor product theorem for crystals and illustrated its applications with concrete examples. In the third chapter we considered Young tableaux and crystals and our main goal was to study the connection between \(U_q(\mathfrak{gl}_n)\)-crystals, which are associated with \(U_q(\mathfrak{gl}_n)\)-modules from category \(\mathcal{O}_{int}^{\geq 0}\) and Young diagrams. More precisely, we demonstrated how the mentioned connection is realized by applying admissible readings to semistandard Young tableaux. Finally, using the obtained realization of the \(U_q(\mathfrak{gl}_n)\)-crystals via Young tableaux, we proved the tensor product decomposition theorem for the corresponding modules. At the end of the chapter, we studied \(U_q(B_3)\)-crystals, gave examples of some crystals of that type and showed some differences that occur in realizations of crystals for Lie algebras of other types.