This PhD thesis consists of an introduction and six papers. All these papers are devoted to Lebesgue norm inequalities with Hardy type integral operators. Three of these papers also deal with so-called Pólya-Knopp type inequalities with geometric mean operators instead of Hardy operators. In the introduction we shortly describe the development and current status of the theory of Hardy type inequalities and put the papers included in this PhD thesis into this frame. The papers are conditionally divided into three parts. The first part consists of three papers, which are devoted to weighted Lebesgue norm inequalities for the Hardy operator with both variable limits of integration. In the first of these papers we characterize this inequality on the cones of non-negative monotone functions with an additional third inner weight function in the definition of the operator. In the second and third papers we find new characterizations for the mentioned inequality and apply the results for characterizing the weighted Lebesgue norm inequality for the corresponding geometric mean operator with both variable limits of integration. The second part consists of two papers, which are connected to operators with monotone kernels. In the first of them we give criteria for boundedness in weighted Lebesgue spaces on the semi-axis of certain integral operators with monotone kernels. In the second one we consider Hardy type inequalities in Lebesgue spaces with general measures for Volterra type integral operators with kernels satisfying some conditions of monotonicity. We establish the equivalence of such inequalities on the cones of non-negative respective non-increasing functions and give some applications. The third part consists of one paper, which is devoted to multi-dimensional Hardy type inequalities. We characterize here some new Hardy type inequalities for operators with Oinarov type kernels and integration over spherical cones in n-dimensional vector space over R. We also obtain some new criteria for a weighted multi-dimensional Hardy inequality (of Sawyer type) to hold with one of two weight functions of product type and give as applications of such results new characterizations of some corresponding n-dimensional weighted Pólya-Knopp inequalities to hold.