The thesis is divided into two sections. Starting with an internal, hyperreal-valued, linear functional $i$ on an internal vector lattice $cal E$ of hyperreal-valued functions on an internal set $Y$, a standard real-valued, linear functional $i^L$ on a standard vector lattice $L$ of real-valued functions on the same internal set $Y$ is developed in the first part of the thesis by using suitable integral norms. Using a special integral norm, called Loeb integral norm, the constructed standardization is the same as the system of Loeb-integrable functions when the functional $i$ is positive. Moreover convergence theorems for the standardization $(L,i^L)$ are proved. Furthermore internal Daniell and Bourbaki integrals are investigated and for any given integral norm, various "local” integral norms are established and their associated convergence structure is analyzed. In the second section of this dissertation the concept of building standardizations with integral norms is generalized. Starting from an internal linear functional $i$, a standard integral $i^L$ with values in the nonstandard hull of an internal vector lattice or of an internal fundamental system is constructed by using suitable integral norms. In particular the standardization of Loeb and Osswald is obtained by using a special integral norm. In addition a monotone convergence theorem is proved.