In this talk, we will introduce the main ideas and concepts of the mathematical framework of Quantum Field Theory. A model is usually defined as a self adjoint operator, called the Hamiltonian, acting on a Hilbert space. The allowed energy values are contained in the spectrum of the Hamiltonian. A first topic that naturally arises is the study of the spectrum of the Hamiltonian. In particular the Hilbert space associated with the absolutely continuous spectrum can be seen as the set of particles which scattered to infinity. Such states play an important role in particle physics experiment and in the so-called scattering theory which we will be presented briefly. However, the predictions of models using physical kernels usually lead to divergences that one has to remove in order to obtain consistent prediction. A procedure, called Renormalisation, is then required. The Goal of this talk is to briefly introduced these concepts, from a functional analysis point of view, which are fundamental aspects of modern Mathematical Physics.