This book is devoted to the contemporary theory of the quantum fields. It is also a comprehensive introduction to modern quantum field theory. The topics under consideration would be of great interest for every physicist and mathematician who are interested in methods of the quantum physics. It is well known that the time evolution of the quantum mechanical system for a single particle is given by the time dependent Schrödinger equation \[ i\hbar { \partial \psi \over\partial t}=H\psi , \] where \(\psi (\mathbf x\)\(,t)\) represents the wave function of the system which corresponds to the probability amplitude for finding the particle at the above coordinate \(\mathbf x\) at a given time \(t\). The Hamiltonian \(H\) is the linear operator \[ H={\mathbf {p }^{2} \over 2m}+V(\mathbf {x}), \] where \(\mathbf {p} =-i\hbar \nabla \) is the operator of the three dimensional momentum to the considered particle, \(m\) is the mass of the particle, and \(V(\mathbf {x})\) represents the potential through which the particle moves. This formalism is clearly non-relativistic and non-covariant. Taking into account the Einstein relation \(E^2=\mathbf{p^2} +m^2\), for a free particle with a mass \(m\), then we obtain the simplest relativistic quantum mechanical equation \(p^{\mu }p_{\mu }\phi =m^2\phi \) (Klein-Gordon equation), where \(p^{\mu }=(E,\mathbf p )\), \(p_{\mu }=(E,-\mathbf p )\) are the energy-momentum four vectors and co-vectors, respectively. Therefore, for particles with vanishing masses it reduces to the wave equation (Maxwell’s equation). In the first chapters the author explains the basic properties characterising the above stated quantum systems. The next chapters are devoted to the solutions of Dirac equation and their properties, representations of Lorentz and Poancaré groups, self-interacting scalar field theory, complex scalar field theory, Maxwell field theory, discrete symmetries and Yang-Mills theory. The covariant quantization of gauge theories (Yang-Mills) is carried out with a detailed description of the BRST symmetry. Feynman diagrams and renormalization theory are given as well. The Higgs phenomenon and the standard model of electroweak interactions are explained systematically. The final chapter is devoted to the renormalization group theory and the renormalization group equation as well as its solving. The \(\phi^4\) theory in four dimension and the Callan - Symanzik equation are considered in the end of the book.