The present work addresses the problems of high-dimensional approximation and uncertainty quantification in the context of electromagnetic field simulations. In the presence of many parameters, one faces the so-called curse of dimensionality. The focus of this work lies on adaptive methods that mitigate the effect of the curse of dimensionality, and therefore enable otherwise intractable uncertainty quantification studies. Its application scope includes electromagnetic field models suffering from moderately high-dimensional input uncertainty. However, the presented methods can be used in a black-box fashion and are therefore applicable to other types of problems as well.