Quantized field theory and the principle of gauge invariance are the foundations of the present successful standard model of elementary particles. In spite of all that has been done there are still some unsolved problems. Two of them are addressed in this book. The first is related to the question of equivalence of a quantized gauge field theory formulated with different gauge conditions. In particular, I show a way how to restore rotational invariance of QCD in the spatial axial gauge and its equivalence with QCD in the Coulomb gauge. The second deals with charged quantum particles in a classical background field. Renormalized effective action describing such interaction is derived by using simple diagrammatic techniques. These incredibly simple ways for calculating multiple-loop Feynman diagrams open the possibility of finding higher orders of Heisenberg-Euler effective action. The book offers a summary of these two topics together with new results and useful references. Knowledge of the basics of quantum field theory is assumed.

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